OFFSET
0,3
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).
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)
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
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 *)
PROG
(PARI) A085537(n) = n^4-n^3
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 05 2003
STATUS
approved