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A001839
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The coding-theoretic function A(n,4,3).
(Formerly M1032 N0389)
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6
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0, 0, 1, 1, 2, 4, 7, 8, 12, 13, 17, 20, 26, 28, 35, 37, 44, 48, 57, 60, 70, 73, 83, 88, 100, 104, 117, 121, 134, 140, 155, 160, 176, 181, 197, 204, 222, 228, 247, 253, 272, 280, 301, 308, 330, 337, 359, 368, 392, 400, 425, 433, 458, 468, 495, 504, 532, 541, 569, 580, 610, 620, 651, 661, 692, 704, 737, 748, 782, 793
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OFFSET
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1,5
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COMMENTS
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Maximum number of edge-disjoint K_3's in a K_n.
Maximum number of clauses in a reduced 1 in 3 SAT instance. Given N items taken three at a time, what is the maximum number of combinations such that no two combinations share more than one item in common. There are reduction rules for 1 in 3 SAT that guarantee no two clauses share more than one variable in common. a(n) is the maximum number of clauses a reduced instance with n variables can have. Example: a(6)=4: (a,b,c)(a,d,e)(b,d,f)(c,e,f). - Russell Easterly, Oct 02 2005
Agrees with independence number of the n-tetrahedral graph for at least a(6)-a(12). - Eric W. Weisstein, Jun 14 2017 and Jul 24 2017
Packing number D(n,3,2). - Rob Pratt, Feb 26 2024
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REFERENCES
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P. J. Cameron, Combinatorics, ..., Cambridge, 1994, see p. 121.
CRC Handbook of Combinatorial Designs, 1996, p. 411.
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).
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LINKS
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R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
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FORMULA
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Known exactly for all n - see Theorem 4 of Brouwer et al. (1990): A(n, 4, 3) = floor((n/3)*floor((n-1)/2))-1 if n is congruent to 5 (mod 6) and A(n, 4, 3) = floor((n/3)*floor((n-1)/2)) if n is not congruent to 5 (mod 6). - Shelly Jones (shellysalt(AT)yahoo.com), Apr 27 2004
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9). - Eric W. Weisstein, Jul 13 2017
G.f.: x^3*(x^5-2*x^4-2*x^3-1) / ((x-1)^3*(x+1)^2*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Sep 21 2013
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EXAMPLE
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Codes illustrating A(4,3,4) = a(3) = 1, A(5,3,4) = a(5) = 2 and A(6,3,4) = a(6) = 4 are:
1110...11100..111000
.......10011..100110
..............010101
..............001011
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MATHEMATICA
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Table[Floor[n Floor[(n - 1)/2]/3] - Boole[Mod[n, 6] == 5], {n, 20}] (* Eric W. Weisstein, Jul 13 2017 *)
Table[(6 n^2 - 9 n - 10 - 3 (-1)^n (n - 2) - 6 Cos[n Pi/3] + 10 Cos[2 n Pi/3] + 10 Sqrt[3] Sin[n Pi/3] + 6 Sqrt[3] Sin[2 n Pi/3])/36, {n, 20}] (* Eric W. Weisstein, Jul 13 2017 *)
LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 0, 1, 1, 2, 4, 7,
CoefficientList[Series[(x^2 (-1 - 2 x^3 - 2 x^4 + x^5))/((-1 + x)^3 (1 + x)^2 (1 - x + x^2) (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 13 2017 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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More terms from Shelly Jones (shellysalt(AT)yahoo.com), Apr 27 2004
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STATUS
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approved
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