cp's OEIS Frontend

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

%I A085537 #72 Feb 16 2025 08:32:50
%S A085537 0,0,8,54,192,500,1080,2058,3584,5832,9000,13310,19008,26364,35672,
%T A085537 47250,61440,78608,99144,123462,152000,185220,223608,267674,317952,
%U A085537 375000,439400,511758,592704,682892,783000,893730,1015808,1149984,1297032,1457750,1632960
%N A085537 a(n) = n^4 - n^3.
%C A085537 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
%C A085537 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
%C A085537 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
%H A085537 Michael B. Porter, <a href="/A085537/b085537.txt">Table of n, a(n) for n = 0..100000</a>
%H A085537 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A085537 Mark L. Lewis, <a href="http://arxiv.org/abs/1206.4334">Bounding group orders by large character degrees: A question of Snyder</a>, arXiv:1206.4334 [math.GR], Jun 19 2012.
%H A085537 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%H A085537 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H A085537 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A085537 From _R. J. Mathar_, Sep 12 2008: (Start)
%F A085537 a(n) = A085540(n-1).
%F A085537 G.f.: 2*x^2*(4 + 7*x + x^2)/(1-x)^5. (End)
%F A085537 a(n) = A000583(n) - A000578(n). - _Omar E. Pol_, Jun 23 2012
%F A085537 Sum_{n>=2} 1/a(n) = 3 - zeta(2) - zeta(3) = A152419. - _Daniel Suteu_, Feb 06 2017
%F A085537 a(n) = 2*A092364(n+1). - _Bruno Berselli_, Sep 08 2017
%F A085537 Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. - _Amiram Eldar_, Jul 05 2020
%F A085537 E.g.f.: exp(x)*x^2*(4 + 5*x + x^2). - _Stefano Spezia_, Jul 06 2021
%F A085537 Product_{n>=2} (1 - 1/a(n)) = A146489. - _Amiram Eldar_, Nov 22 2022
%t A085537 Table[(n - 1) n^3, {n, 0, 20}] (* _Eric W. Weisstein_, Sep 08 2017 *)
%t A085537 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 54, 192, 500}, {0, 20}] (* _Eric W. Weisstein_, Sep 08 2017 *)
%t A085537 CoefficientList[Series[2 x^2 (4 + 7 x + x^2)/(1 - x)^5, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 08 2017 *)
%o A085537 (PARI) A085537(n) = n^4-n^3
%Y A085537 A diagonal of A228273.
%Y A085537 Cf. A000578, A000583, A092364, A146489.
%Y A085537 Cf. A085540 (same sequence with initial 0 dropped).
%K A085537 nonn,easy
%O A085537 0,3
%A A085537 _N. J. A. Sloane_, Jul 05 2003