OFFSET
0,6
COMMENTS
a(n) is the maximum number of 4-circuits in an n-tournament. - Franck Maminirina Ramaharo, Aug 06 2025
REFERENCES
K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 186, Theorem 6.11.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).
FORMULA
From Colin Barker, Nov 19 2016: (Start)
a(n) = (n^4-3*n^3-4*n^2+12*n)/48 for n even.
a(n) = (n^4-3*n^3-n^2+3*n)/48 for n odd.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n>7.
G.f.: x^4*(1+3*x) / ((1-x)^5 * (1+x)^3). (End)
a(n) = n*(n - 3)*(2*n^2 - 3*(-1)^n - 5)/96. - Paolo Xausa, Sep 17 2024 (derived from Bruno Berselli formula in A006918)
From Amiram Eldar, Dec 04 2025: (Start)
Sum_{n>=4} 1/a(n) = 96*log(2)/5 - 179/15.
Sum_{n>=4} (-1)^n/a(n) = 146/15 - 64*log(2)/5. (End)
E.g.f.: x*(x*(x^2 + 3*x - 6)*cosh(x) + (x^3 + 3*x^2 - 3*x + 6)*sinh(x))/48. - Stefano Spezia, Dec 14 2025
MATHEMATICA
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 0, 0, 1, 5, 12, 28}, 100]
(* or *)
A038376[n_] := n*(n - 3)*(2*n^2 - 3*(-1)^n - 5)/96;
Array[A038376, 100, 0] (* Paolo Xausa, Sep 16 2024 *)
PROG
(PARI) concat(vector(4), Vec(x^4*(1+3*x) / ((1-x)^5 * (1+x)^3) + O(x^100))) \\ Colin Barker, Nov 19 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Name corrected by Paolo Xausa, Sep 16 2024
STATUS
approved
