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Search: a028594 -id:a028594
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A205976 a(n) = Fibonacci(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2. +20
3
1, 4, 12, 32, 84, 120, 384, 52, 1260, 1768, 3960, 4272, 16128, 13048, 4524, 58560, 122388, 114984, 403104, 334480, 1136520, 175136, 2550384, 2751072, 11128320, 9303100, 20394024, 31426880, 8898708, 61707480, 239627520, 172322432, 548933868, 676718976, 1231823592 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A028594:
1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).
Here Chi(n,7) = principal Dirichlet character of n modulo 7.
LINKS
Table of n, a(n) for n=0..34.
FORMULA
G.f.: 1 + 4*Sum_{n>=1} Fibonacci(n)*Chi(n,7)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
EXAMPLE
G.f.: A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + 84*x^4 + 120*x^5 + 384*x^6 + 52*x^7 +...
where A(x) = 1 + 1*4*x + 1*12*x^2 + 2*16*x^3 + 3*28*x^4 + 5*24*x^5 + 8*48*x^6 + 13*4*x^7 + 21*60*x^8 + 34*52*x^9 +...+ Fibonacci(n)*A028594(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1-18*x^6+x^12) + 0*13*7*x^7/(1+29*x^7-x^14) +...).
The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].
PROG
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 4*sum(m=1, n, fibonacci(m)*kronecker(m, 7)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", "))
CROSSREFS
Cf. A028594, A205975, A203847, A000204 (Lucas).
Cf. A209456 (Pell variant).
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2012
STATUS
approved
A209456 a(n) = Pell(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2. +20
3
1, 4, 24, 80, 336, 696, 3360, 676, 24480, 51220, 171216, 275568, 1552320, 1873816, 969384, 18722400, 58383168, 81841608, 428096760, 530008720, 2687063904, 617823440, 13424019552, 21605633376, 130401532800, 162655527004, 532025081616, 1223259207200 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A028594:
1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).
Here Chi(n,7) = principal Dirichlet character of n modulo 7.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1 + 4*Sum_{n>=1} Pell(n)*Chi(n,7)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
EXAMPLE
G.f.: A(x) = 1 + 4*x + 24*x^2 + 80*x^3 + 336*x^4 + 696*x^5 + 3360*x^6 +...
where A(x) = 1 + 1*4*x + 2*12*x^2 + 5*16*x^3 + 12*28*x^4 + 29*24*x^5 + 70*48*x^6 + 169*4*x^7 + 408*60*x^8 + 985*52*x^9 +...+ Pell(n)*A028594(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 5*3*x^3/(1-14*x^3-x^6) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 70*6*x^6/(1-198*x^6+x^12) + 0*169*7*x^7/(1+478*x^7-x^14) +...).
The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].
MATHEMATICA
A028594[n_]:= If[n < 1, Boole[n == 0], 4*Sum[If[Mod[d, 7] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A028594[n], {n, 1, 50}]] (* G. C. Greubel, Jan 03 2018 *)
PROG
(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 + 4*sum(m=1, n, Pell(m)*kronecker(m, 7)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 35, print1(a(n), ", "))
CROSSREFS
Cf. A028594, A205976, A209455, A204270, A000129 (Pell), A002203.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 10 2012
STATUS
approved
A113957 Sum of the divisors of n which are not divisible by 7. +10
13
1, 3, 4, 7, 6, 12, 1, 15, 13, 18, 12, 28, 14, 3, 24, 31, 18, 39, 20, 42, 4, 36, 24, 60, 31, 42, 40, 7, 30, 72, 32, 63, 48, 54, 6, 91, 38, 60, 56, 90, 42, 12, 44, 84, 78, 72, 48, 124, 1, 93, 72, 98, 54, 120, 72, 15, 80, 90, 60, 168, 62, 96, 13, 127, 84, 144, 68, 126, 96, 18, 72 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 467, Entry 5(i).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) is multiplicative and a(p^e) = 1, if p=7, a(p^e) = (p^(e+1)-1)/(p-1) otherwise.
G.f.: ((theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^2-1)/4.
L.g.f.: log(Product_{k>=1} (1 - x^(7*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
Sum_{k=1..n} a(k) ~ (Pi^2/14) * n^2. - Amiram Eldar, Oct 04 2022
Dirichlet g.f. (1-7^(1-s))*zeta(s)*zeta(s-1). - R. J. Mathar, May 17 2023
MATHEMATICA
f[p_, e_] := If[p == 7, 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)
PROG
(PARI) a(n)=if(n<1, 0, sigma(n/7^valuation(n, 7)))
CROSSREFS
Cf. A028594(n)=4*a(n) if n>0.
Cf. A244600.
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Nov 10 2005
STATUS
approved
A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1. +10
5
1, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 6, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 18, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 24, 140 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to sums of divisors.
FORMULA
a(n) is multiplicative with a(0) = 1, a(5^e) = 6 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = 25 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + Sum_{k>0} k * x^k / (1 - x^k) - Sum_{k>0} 25 * k * x^(25*k) / (1 - x^(25*k)).
Sum_{k=1..n} a(k) ~ (2*Pi^2/25) * n^2. - Amiram Eldar, Oct 04 2022
EXAMPLE
G.f. = 1 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + ...
75 has six divisors: 1, 3, 5, 15, 25, 75, but both 25 and 75 are divisible by 25, thus not counted, and we have a(75) = 1+3+5+15 = 24. - Antti Karttunen, Nov 23 2017
MATHEMATICA
a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ If[ Mod[ d, 25] > 0, d, 0], {d, Divisors @ n}]];
PROG
(PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, if( d%25, d)))};
(PARI) {a(n) = if( n<1, n==0, 1 * (sigma(n) - if( n%25==0, 25 * sigma( n / 25))))};
(Sage) A = ModularForms( Gamma0(25), 2, prec=66) . basis(); A[0] + A[1] + 3*A[2] + 4*A[3] + 7*A[4];
(Magma) A := Basis( ModularForms( Gamma0(25), 2), 66); A[1] + A[2] + 3*A[3] + 4*A[4] + 7*A[5]; /* Michael Somos, Jun 12 2014 */
CROSSREFS
Cf. A000118, A004011, A008653, A028594, A028887, A096726, A107505.
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Jul 02 2013
EXTENSIONS
More terms from Antti Karttunen, Nov 23 2017
STATUS
approved
A366091 a(n) is the number of ways to write n = i^2 + 2*j^2 + 3*k^2 with i,j,k >= 0. +10
3
1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 0, 2, 4, 1, 2, 2, 2, 1, 3, 2, 2, 4, 2, 1, 2, 2, 0, 4, 3, 2, 5, 2, 1, 3, 2, 2, 7, 2, 2, 5, 0, 2, 0, 2, 4, 4, 3, 1, 4, 3, 3, 5, 3, 2, 7, 1, 2, 6, 0, 3, 6, 2, 2, 4, 2, 2, 6, 3, 2, 4, 3, 3, 3, 2, 0, 7, 5, 2, 6, 3, 2, 8, 2, 2, 11, 2, 5, 2, 2, 3, 0, 4, 3, 7, 3, 2, 2, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
FORMULA
G.f. (1 + theta_3(0,z)) * (1 + theta_3(0,z^2)) * (1 + theta_3(0,z^3))/8 where theta_3 is a Jacobi theta function.
EXAMPLE
a(9) = 3 because 9 = 3^2 + 2*0^2 + 3*0^2 = 1^2 + 2*2^2 + 3*0^2 = 2^2 + 2*1^2 + 3*1^2.
MAPLE
g:= (1+JacobiTheta3(0, z))*(1+JacobiTheta3(0, z^2))*(1+JacobiTheta3(0, z^3))/8:
S:= series(g, z, 101):
seq(coeff(S, z, j), j=0..100);
PROG
(Python)
from itertools import count
from sympy.ntheory.primetest import is_square
def A366091(n):
c = 0
for k in count(0):
if (a:=3*k**2)>n:
break
for j in count(0):
if (b:=a+(j**2<<1))>n:
break
if is_square(n-b):
c += 1
return c # Chai Wah Wu, Sep 29 2023
CROSSREFS
Cf. A028594 (allows any integer i,j,k), A055042 (a(n) = 0)
KEYWORD
nonn
AUTHOR
Robert Israel, Sep 28 2023
STATUS
approved
A028596 Expansion of (theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^4. +10
1
1, 8, 40, 128, 328, 656, 1216, 1864, 2856, 3560, 5392, 6368, 9856, 10640, 17000, 16832, 22600, 23760, 32776, 32576, 43792, 52864, 57568, 58560, 78528, 76024, 94864, 98432, 137864, 116720, 152512, 143040, 179240, 179072, 212112, 237328, 265768, 242352, 296704, 295232 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Jinyuan Wang, Table of n, a(n) for n = 0..1000
PROG
(PARI) a(n) = polcoeff((1 + 2*x*Ser(qfrep([2, 1; 1, 4], n, 1)))^4, n); \\ Jinyuan Wang, Feb 21 2020
CROSSREFS
Cf. A002652, A002653, A028594.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Jinyuan Wang, Feb 21 2020
STATUS
approved
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Last modified September 13 09:59 EDT 2024. Contains 375904 sequences. (Running on oeis4.)