A038376 a(n) = (n-3)*A006918(n-2)/2 for n >= 2, with a(0) = a(1) = 0.

0, 0, 0, 0, 1, 5, 12, 28, 50, 90, 140, 220, 315, 455, 616, 840, 1092, 1428, 1800, 2280, 2805, 3465, 4180, 5060, 6006, 7150, 8372, 9828, 11375, 13195, 15120, 17360, 19720, 22440, 25296, 28560, 31977, 35853, 39900, 44460, 49210, 54530, 60060, 66220, 72611

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A006918.

Programs

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 *)

PARI

concat(vector(4), Vec(x^4*(1+3*x) / ((1-x)^5 * (1+x)^3) + O(x^100))) \\ Colin Barker, Nov 19 2016

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)

Extensions

Name corrected by Paolo Xausa, Sep 16 2024