A085537 a(n) = n^4 - n^3.

0, 0, 8, 54, 192, 500, 1080, 2058, 3584, 5832, 9000, 13310, 19008, 26364, 35672, 47250, 61440, 78608, 99144, 123462, 152000, 185220, 223608, 267674, 317952, 375000, 439400, 511758, 592704, 682892, 783000, 893730, 1015808, 1149984, 1297032, 1457750, 1632960

Offset: 0

Author

N. J. A. Sloane, Jul 05 2003

Keywords

• nonn
• easy

Comments

For n>=1, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for a fixed x in {1,2,3,4} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007

Let K_n denote the complete graph on n (n>1) vertices. The sequence corresponds to the Wiener index of K_n X K_n (Cartesian product of K_n with itself). - K.V.Iyer, Mar 12 2009

Lewis proved that the order of a solvable nonabelian finite group |G| is less than or equal to e^4 - e^3, where when d is an irreducible character degree of G, then there is a positive integer e such that |G| = d(d+e). - Jonathan Vos Post, Jun 21 2012

Links

• Michael B. Porter, Table of n, a(n) for n = 0..100000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• Mark L. Lewis, Bounding group orders by large character degrees: A question of Snyder, arXiv:1206.4334 [math.GR], Jun 19 2012.
• Eric Weisstein's World of Mathematics, Rook Graph.
• Eric Weisstein's World of Mathematics, Wiener Index.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

A diagonal of A228273.

Cf. A000578, A000583, A092364, A146489.

Cf. A085540 (same sequence with initial 0 dropped).

Programs

Mathematica

Table[(n - 1) n^3, {n, 0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 54, 192, 500}, {0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[2 x^2 (4 + 7 x + x^2)/(1 - x)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

PARI

A085537(n) = n^4-n^3

Formula

From R. J. Mathar, Sep 12 2008: (Start)

a(n) = A085540(n-1).

G.f.: 2*x^2*(4 + 7*x + x^2)/(1-x)^5. (End)

a(n) = A000583(n) - A000578(n). - Omar E. Pol, Jun 23 2012

Sum_{n>=2} 1/a(n) = 3 - zeta(2) - zeta(3) = A152419. - Daniel Suteu, Feb 06 2017

a(n) = 2*A092364(n+1). - Bruno Berselli, Sep 08 2017

Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. - Amiram Eldar, Jul 05 2020

E.g.f.: exp(x)*x^2*(4 + 5*x + x^2). - Stefano Spezia, Jul 06 2021

Product_{n>=2} (1 - 1/a(n)) = A146489. - Amiram Eldar, Nov 22 2022