Search: a015445 -id:a015445
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A109466
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Riordan array (1, x(1-x)).
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+10
53
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1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 1, -3, 1, 0, 0, 0, 3, -4, 1, 0, 0, 0, -1, 6, -5, 1, 0, 0, 0, 0, -4, 10, -6, 1, 0, 0, 0, 0, 1, -10, 15, -7, 1, 0, 0, 0, 0, 0, 5, -20, 21, -8, 1, 0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1, 0, 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1, 0, 0, 0, 0, 0, 0, 1, -21, 70, -84, 45, -11, 1, 0, 0, 0, 0
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OFFSET
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0,9
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COMMENTS
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Inverse is Riordan array (1, xc(x)) (A106566).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Coefficient array of the polynomials Chebyshev_U(n, sqrt(x)/2)*(sqrt(x))^n. - Paul Barry, Sep 28 2009
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LINKS
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FORMULA
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Number triangle T(n, k) = (-1)^(n-k)*binomial(k, n-k).
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = -12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. - Philippe Deléham, Oct 27 2008
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively. - Philippe Deléham, Oct 28 2008
Sum_{k=0..n} T(n,k)*x^(n-k) = F(n+1,-x) where F(n,x)is the n-th Fibonacci polynomial in x defined in A011973. - Philippe Deléham, Feb 22 2013
For T(0,0) = 0, the signed triangle below has the o.g.f. G(x,t) = [t*x(1-x)]/[1-t*x(1-x)] = L[t*Cinv(x)] where L(x) = x/(1-x) and Cinv(x)=x(1-x) with the inverses Linv(x) = x/(1+x) and C(x)= [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108, so the inverse o.g.f. is Ginv(x,t) = C[Linv(x)/t] = [1-sqrt[1-4*x/(t(1+x))]]/2 (cf. A124644 and A030528). - Tom Copeland, Jan 19 2016
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EXAMPLE
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Rows begin:
1;
0, 1;
0, -1, 1;
0, 0, -2, 1;
0, 0, 1, -3, 1;
0, 0, 0, 3, -4, 1;
0, 0, 0, -1, 6, -5, 1;
0, 0, 0, 0, -4, 10, -6, 1;
0, 0, 0, 0, 1, -10, 15, -7, 1;
0, 0, 0, 0, 0, 5, -20, 21, -8, 1;
0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1;
Production array is
0, 1,
0, -1, 1,
0, -1, -1, 1,
0, -2, -1, -1, 1,
0, -5, -2, -1, -1, 1,
0, -14, -5, -2, -1, -1, 1,
0, -42, -14, -5, -2, -1, -1, 1,
0, -132, -42, -14, -5, -2, -1, -1, 1,
0, -429, -132, -42, -14, -5, -2, -1, -1, 1 (End)
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MATHEMATICA
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(* The function RiordanArray is defined in A256893. *)
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PROG
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(Magma) /* As triangle */ [[(-1)^(n-k)*Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 14 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A168561
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Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.
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+10
28
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1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 1, 0, 6, 0, 5, 0, 1, 0, 4, 0, 10, 0, 6, 0, 1, 1, 0, 10, 0, 15, 0, 7, 0, 1, 0, 5, 0, 20, 0, 21, 0, 8, 0, 1, 1, 0, 15, 0, 35, 0, 28, 0, 9, 0, 1, 0, 6, 0, 35, 0, 56, 0, 36, 0, 10, 0, 1, 1, 0, 21, 0, 70, 0, 84, 0, 45, 0, 11, 0, 1
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OFFSET
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0,8
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COMMENTS
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Row sums: A000045(n+1), Fibonacci numbers.
T(n,k) is the number of compositions of n+1 into k+1 odd parts. Example: T(4,2)=3 because we have 5 = 1+1+3 = 1+3+1 = 3+1+1.
Coefficients of monic Fibonacci polynomials (rising powers of x). Ftilde(n, x) = x*Ftilde(n-1, x) + Ftilde(n-2, x), n >=0, Ftilde(-1,x) = 0, Ftilde(0, x) = 1. G.f.: 1/(1 - x*z - z^2). Compare with Chebyshev S-polynomials (A049310). - Wolfdieter Lang, Jul 29 2014
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LINKS
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FORMULA
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Sum_{k=0..n} T(n,k)*x^k = A059841(n), A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 respectively. - Philippe Deléham, Dec 02 2009
T(2n,2k) = A085478(n,k). T(2n+1,2k+1) = A078812(n,k). Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000045(n+1), A006131(n), A015445(n), A168579(n), A122999(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Dec 02 2009
T(n,k) = binomial((n+k)/2,k) if (n+k) is even; otherwise T(n,k)=0.
G.f.: (1-z^2)/(1-t*z-z^2) if offset is 1.
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = 1, T(0,1) = 0. - Philippe Deléham, Feb 09 2012
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EXAMPLE
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The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
0: 1
1: 0 1
2: 1 0 1
3: 0 2 0 1
4: 1 0 3 0 1
5: 0 3 0 4 0 1
6: 1 0 6 0 5 0 1
7: 0 4 0 10 0 6 0 1
8: 1 0 10 0 15 0 7 0 1
9: 0 5 0 20 0 21 0 8 0 1
10: 1 0 15 0 35 0 28 0 9 0 1
11: 0 6 0 35 0 56 0 36 0 10 0 1
12: 1 0 21 0 70 0 84 0 45 0 11 0 1
13: 0 7 0 56 0 126 0 120 0 55 0 12 0 1
14: 1 0 28 0 126 0 210 0 165 0 66 0 13 0 1
15: 0 8 0 84 0 252 0 330 0 220 0 78 0 14 0 1
------------------------------------------------------------------------
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MAPLE
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A168561:=proc(n, k) if n-k mod 2 = 0 then binomial((n+k)/2, k) else 0 fi end proc:
seq(seq(A168561(n, k), k=0..n), n=0..12) ; # yields sequence in triangular form
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MATHEMATICA
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Table[If[EvenQ[n + k], Binomial[(n + k)/2, k], 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 16 2017 *)
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PROG
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(PARI) T(n, k) = if ((n+k) % 2, 0, binomial((n+k)/2, k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print(); ); \\ Michel Marcus, Oct 09 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Typo in name corrected (1(1-x^2) changed to 1/(1-x^2)) by Wolfdieter Lang, Nov 20 2010
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STATUS
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approved
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A002534
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a(n) = 2*a(n-1) + 9*a(n-2), with a(0) = 0, a(1) = 1.
(Formerly M2058 N0814)
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+10
21
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0, 1, 2, 13, 44, 205, 806, 3457, 14168, 59449, 246410, 1027861, 4273412, 17797573, 74055854, 308289865, 1283082416, 5340773617, 22229288978, 92525540509, 385114681820, 1602959228221, 6671950592822, 27770534239633, 115588623814664
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OFFSET
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0,3
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COMMENTS
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For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 2's along the main diagonal, and 3's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 19 2011
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
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LINKS
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FORMULA
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E.g.f.: exp(x)*sinh(sqrt(10)*x)/sqrt(10).
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*10^k. (End)
a(n) = ((1+sqrt(10))^n - (1-sqrt(10))^n)/(2*sqrt(10)). - Artur Jasinski, Dec 10 2006
G.f.: x/(1 - 2*x - 9*x^2) - Iain Fox, Jan 17 2018
a(n) = (3*i)^(n-1)*ChebyshevU(n-1, -i/3).
a(n) = 3^(n-1)*Fibonacci(n, 2/3), where Fibonacci(n, x) is the Fibonacci polynomial. (End)
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MAPLE
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MATHEMATICA
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Table[((1 + Sqrt[10])^n - (1 - Sqrt[10])^n)/(2 Sqrt[10]), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *)
LinearRecurrence[{2, 9}, {0, 1}, 30] (* T. D. Noe, Aug 18 2011 *)
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PROG
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(Sage) [lucas_number1(n, 2, -9) for n in range(0, 20)] # Zerinvary Lajos, Apr 22 2009
(Magma) [Ceiling(((1+Sqrt(10))^n-(1-Sqrt(10))^n)/(2*Sqrt(10))): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011
(PARI) first(n) = Vec(x/(1 - 2*x - 9*x^2) + O(x^n), -n) \\ Iain Fox, Jan 17 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A057089
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Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.
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+10
15
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1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440
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OFFSET
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0,2
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COMMENTS
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a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's of this word is 6*a(n-1) and 6*a(n-2), resp.
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LINKS
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FORMULA
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a(n) = 6*a(n-1) + 6*a(n-2); a(0)=1, a(1)=6.
a(n) = S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-6*x-6*x^2).
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MATHEMATICA
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LinearRecurrence[{6, 6}, {1, 6}, 40] (* Harvey P. Dale, Nov 05 2011 *)
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PROG
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(Sage) [lucas_number1(n, 6, -6) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009
(Magma) I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)+6*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
(PARI) x='x+O('x^30); Vec(1/(1-6*x-6*x^2)) \\ G. C. Greubel, Jan 24 2018
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CROSSREFS
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Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144
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OFFSET
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0,6
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COMMENTS
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Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
Triangle read by rows: T(n,n-k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)). - Philippe Deléham, Feb 17 2014
T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014
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LINKS
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FORMULA
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T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 1, 2;
0, 0, 2, 3;
0, 0, 1, 5, 5;
0, 0, 0, 3, 10, 8;
0, 0, 0, 1, 9, 20, 13;
0, 0, 0, 0, 4, 22, 38, 21;
0, 0, 0, 0, 1, 14, 51, 71, 34;
0, 0, 0, 0, 0, 5, 40, 111, 130, 55;
0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89;
0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144;
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PROG
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(Haskell)
a111006 n k = a111006_tabl !! n !! k
a111006_row n = a111006_tabl !! n
a111006_tabl = map fst $ iterate (\(us, vs) ->
(vs, zipWith (+) (zipWith (+) ([0] ++ us ++ [0]) ([0, 0] ++ us))
([0] ++ vs))) ([1], [0, 1])
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A135030
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Generalized Fibonacci numbers: a(n) = 6*a(n-1) + 2*a(n-2).
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+10
9
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0, 1, 6, 38, 240, 1516, 9576, 60488, 382080, 2413456, 15244896, 96296288, 608267520, 3842197696, 24269721216, 153302722688, 968355778560, 6116740116736, 38637152257536, 244056393778688, 1541612667187200
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OFFSET
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0,3
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COMMENTS
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For n>0, a(n) equals the number of words of length n-1 over {0,1,...,7} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017
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LINKS
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FORMULA
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a(0) = 0; a(1) = 1; a(n) = 2*(3*a(n-1) + a(n-2)).
a(n) = 1/(2*sqrt(11))*( (3 + sqrt(11))^n - (3 - sqrt(11))^n ).
E.g.f.: (1/sqrt(11))*exp(3*x)*sinh(sqrt(11)*x). - G. C. Greubel, Sep 17 2016
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MAPLE
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A:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(n) = 2*(3*a(n-1) + a(n-2))}, a(n), remember):
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MATHEMATICA
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LinearRecurrence[{6, 2}, {0, 1}, 30] (* or *) CoefficientList[Series[ -(x/(2x^2+6x-1)), {x, 0, 30}], x] (* Harvey P. Dale, Jun 20 2011 *)
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PROG
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(Sage) [lucas_number1(n, 6, -2) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009
(Magma) [n le 2 select n-1 else 6*Self(n-1) + 2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Sep 18 2016
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CROSSREFS
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Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A180250.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A083856
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Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).
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+10
8
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0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 5, 5, 1, 0, 1, 1, 5, 7, 11, 8, 1, 0, 1, 1, 6, 9, 19, 21, 13, 1, 0, 1, 1, 7, 11, 29, 40, 43, 21, 1, 0, 1, 1, 8, 13, 41, 65, 97, 85, 34, 1, 0, 1, 1, 9, 15, 55, 96, 181, 217, 171, 55, 1
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OFFSET
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0,14
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COMMENTS
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Row n >= 0 of the array gives the solution to the recurrence b(k) = b(k-1) + n*b(k-2) for k >= 2 with b(0) = 0 and b(1) = 1.
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LINKS
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FORMULA
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T(n, k) = (((1 + sqrt(4*n + 1))/2)^k - ((1 - sqrt(4*n + 1))/2)^k)/sqrt(4*n + 1). [corrected by Michel Marcus, Jun 25 2018]
The g.f. for row n >= 0 is x/(1 - x - n*x^2).
The g.f. for column k >= 1 is g(k,x) = 1/(1-x) + Sum_{m = 1..floor((k-1)/2)} (1 - x)^(-1 - m) * binomial(k - 1 - m, m) * Sum_{i = 0..m} x^i * Sum_{j = 0..i} (-1)^j * (i - j)^m * binomial(1 + m, j).
The g.f. for column k >= 1 is also g(k,x) = 1 + Sum_{m = 1..floor((k+1)/2)} ((1 - x)^(-m) * binomial(k-m, m-1) * Sum_{j = 0..m} (-1)^j * binomial(m, j) * x^m * Phi(x, -m+1, -j+m)) + Sum_{s = 0..floor((k-1)/2)} binomial(k-s-1, s) * PolyLog(-s, x), where Phi is the Lerch transcendent function. (End)
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EXAMPLE
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Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... [A057427]
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... [A000045]
0, 1, 1, 3, 5, 11, 21, 43, 85, 171, ... [A001045]
0, 1, 1, 4, 7, 19, 40, 97, 217, 508, ... [A006130]
0, 1, 1, 5, 9, 29, 65, 181, 441, 1165, ... [A006131]
0, 1, 1, 6, 11, 41, 96, 301, 781, 2286, ... [A015440]
0, 1, 1, 7, 13, 55, 133, 463, 1261, 4039, ... [A015441]
0, 1, 1, 8, 15, 71, 176, 673, 1905, 6616, ... [A015442]
0, 1, 1, 9, 17, 89, 225, 937, 2737, 10233, ... [A015443]
0, 1, 1, 10, 19, 109, 280, 1261, 3781, 15130, ... [A015445]
...
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MAPLE
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A083856_row := proc(r, n) local R; R := proc(n) option remember;
if n<=1 then n else R(n-1)+r*R(n-2) fi end: R(n) end:
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MATHEMATICA
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T[_, 0] = 0; T[_, 1|2] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
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PROG
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(Julia)
function generalized_fibonacci(r, n)
F = BigInt[1 r; 1 0]
Fn = F^n
Fn[2, 1]
end
for r in 0:6 println([generalized_fibonacci(r, n) for n in 0:9]) end # Peter Luschny, Mar 06 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A180250
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a(n) = 5*a(n-1) + 10*a(n-2), with a(1)=0 and a(2)=1.
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+10
7
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0, 1, 5, 35, 225, 1475, 9625, 62875, 410625, 2681875, 17515625, 114396875, 747140625, 4879671875, 31869765625, 208145546875, 1359425390625, 8878582421875, 57987166015625, 378721654296875, 2473479931640625, 16154616201171875, 105507880322265625
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = ((5+sqrt(65))^(n-1) - (5-sqrt(65))^(n-1))/(2^(n-1)*sqrt(65)). - Rolf Pleisch, May 14 2011
G.f.: x^2/(1-5*x-10*x^2).
a(n) = (i*sqrt(10))^(n-1) * ChebyshevU(n-1, -i*sqrt(5/8)). - G. C. Greubel, Jul 21 2023
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MATHEMATICA
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Join[{a=0, b=1}, Table[c=5*b+10*a; a=b; b=c, {n, 100}]]
LinearRecurrence[{5, 10}, {0, 1}, 30] (* G. C. Greubel, Jan 16 2018 *)
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PROG
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(PARI) my(x='x+O('x^30)); concat([0], Vec(x^2/(1-5*x-10*x^2))) \\ G. C. Greubel, Jan 16 2018
(Magma) [n le 2 select n-1 else 5*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018
(SageMath)
A180250= BinaryRecurrenceSequence(5, 10, 0, 1)
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CROSSREFS
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Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A015551
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Expansion of x/(1 - 6*x - 5*x^2).
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+10
6
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0, 1, 6, 41, 276, 1861, 12546, 84581, 570216, 3844201, 25916286, 174718721, 1177893756, 7940956141, 53535205626, 360916014461, 2433172114896, 16403612761681, 110587537144566, 745543286675801, 5026197405777636
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graph;
refs;
listen;
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text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Let the generator matrix for the ternary Golay G_12 code be [I|B], where the elements of B are taken from the set {0,1,2}. Then a(n)=(B^n)_1,2 for instance. - Paul Barry, Feb 13 2004
Pisano period lengths: 1, 2, 4, 4, 1, 4, 42, 8, 12, 2, 10, 4, 12, 42, 4, 16, 96, 12, 360, 4, ... - R. J. Mathar, Aug 10 2012
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LINKS
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FORMULA
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a(n) = 6*a(n-1) + 5*a(n-2).
a(n) = sqrt(14)*(3+sqrt(14))^n/28 - sqrt(14)*(3-sqrt(14))^n/28. - Paul Barry, Feb 13 2004
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MATHEMATICA
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CoefficientList[Series[x/(1-6x-5x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[ {6, 5}, {0, 1}, 30] (* Harvey P. Dale, Oct 30 2017 *)
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PROG
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(Sage) [lucas_number1(n, 6, -5) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009
(Magma) I:=[0, 1]; [n le 2 select I[n] else 6*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
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CROSSREFS
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Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A057089, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A193376
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T(n,k) = number of ways to place any number of 2 X 1 tiles of k distinguishable colors into an n X 1 grid; array read by descending antidiagonals, with n, k >= 1.
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+10
6
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1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 11, 8, 1, 6, 9, 19, 21, 13, 1, 7, 11, 29, 40, 43, 21, 1, 8, 13, 41, 65, 97, 85, 34, 1, 9, 15, 55, 96, 181, 217, 171, 55, 1, 10, 17, 71, 133, 301, 441, 508, 341, 89, 1, 11, 19, 89, 176, 463, 781, 1165, 1159, 683, 144, 1, 12, 21, 109, 225, 673
(list;
table;
graph;
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history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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As to the sequences by columns beginning (1, N, ...), let m = (N-1). The g.f. for the sequence (1, N, ...) is 1/(1 - x - m*x^2). Alternatively, the corresponding matrix generator is [[1,1], [m,0]]. Another equivalency is simply: The sequence beginning (1, N, ...) is the INVERT transform of (1, m, 0, 0, 0, ...). Convergents to the sequences a(n)/a(n-1) are (1 + sqrt(4*m+1))/2. - Gary W. Adamson, Feb 25 2014
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LINKS
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FORMULA
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With z X 1 tiles of k colors on an n X 1 grid (with n >= z), either there is a tile (of any of the k colors) on the first spot, followed by any configuration on the remaining (n-z) X 1 grid, or the first spot is vacant, followed by any configuration on the remaining (n-1) X 1. Thus, T(n,k) = T(n-1,k) + k*T(n-z,k), with T(n,k) = 1 for n = 0, 1, ..., z-1.
The solution is T(n,k) = Sum_r r^(-n-1)/(1 + z*k*r^(z-1)), where the sum is over the roots r of the polynomial k*x^z + x - 1.
For z = 2, T(n,k) = ((2*k / (sqrt(1 + 4*k) - 1))^(n+1) - (-2*k/(sqrt(1 + 4*k) + 1))^(n+1)) / sqrt(1 + 4*k).
T(n,k) = Sum_{s=0..[n/2]} binomial(n-s,s) * k^s.
For z X 1 tiles, T(n,k,z) = Sum_{s = 0..[n/z]} binomial(n-(z-1)*s, s) * k^s. - R. H. Hardin, Jul 31 2011
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EXAMPLE
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Array T(n,k) (with rows n >= 1 and column k >= 1) begins as follows:
..1...1....1....1.....1.....1.....1......1......1......1......1......1...
..2...3....4....5.....6.....7.....8......9.....10.....11.....12.....13...
..3...5....7....9....11....13....15.....17.....19.....21.....23.....25...
..5..11...19...29....41....55....71.....89....109....131....155....181...
..8..21...40...65....96...133...176....225....280....341....408....481...
.13..43...97..181...301...463...673....937...1261...1651...2113...2653...
.21..85..217..441...781..1261..1905...2737...3781...5061...6601...8425...
.34.171..508.1165..2286..4039..6616..10233..15130..21571..29844..40261...
.55.341.1159.2929..6191.11605.19951..32129..49159..72181.102455.141361...
.89.683.2683.7589.17621.35839.66263.113993.185329.287891.430739.624493...
...
Some solutions for n = 5 and k = 3 with colors = 1, 2, 3 and empty = 0:
..0....2....3....2....0....1....0....0....2....0....0....2....3....0....0....0
..0....2....3....2....2....1....2....3....2....1....0....2....3....1....1....1
..1....0....0....0....2....0....2....3....2....1....0....1....0....1....1....1
..1....2....2....0....3....2....2....3....2....0....3....1....3....3....2....1
..0....2....2....0....3....2....2....3....0....0....3....0....3....3....2....1
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MAPLE
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T:= proc(n, k) option remember; `if`(n<0, 0,
`if`(n<2 or k=0, 1, k*T(n-2, k) +T(n-1, k)))
end;
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[n < 0, 0, If[n < 2 || k == 0, 1, k*T[n-2, k]+T[n-1, k]]]; Table[Table[T[n, d+1-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Mar 04 2014, after Alois P. Heinz *)
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CROSSREFS
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Column 1 is A000045(n+1), column 2 is A001045(n+1), column 3 is A006130, column 4 is A006131, column 5 is A015440, column 6 is A015441(n+1), column 7 is A015442(n+1), column 8 is A015443, column 9 is A015445, column 10 is A015446, column 11 is A015447, and column 12 is A053404,
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KEYWORD
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AUTHOR
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EXTENSIONS
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Formula and proof from Robert Israel in the Sequence Fans mailing list.
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STATUS
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approved
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