A085490 Number of pairs with two different elements which can be obtained by selecting unique elements from two sets with n+1 and n^2 elements respectively and n common elements.

0, 1, 10, 33, 76, 145, 246, 385, 568, 801, 1090, 1441, 1860, 2353, 2926, 3585, 4336, 5185, 6138, 7201, 8380, 9681, 11110, 12673, 14376, 16225, 18226, 20385, 22708, 25201, 27870, 30721, 33760, 36993, 40426, 44065, 47916, 51985, 56278, 60801, 65560, 70561, 75810, 81313

Offset: 0

Author

Polina S. Dolmatova (polinasport(AT)mail.ru), Aug 15 2003

Keywords

• nonn
• easy

Examples

			a(2) = 10 because we can write a(2) = 2^3 + 2^2 - 2 = 10.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A270109.

Programs

Magma

[n^3+n^2-n: n in [0..50]]; // Vincenzo Librandi, Jun 22 2017

Maple

a:=n->sum(n*k, k=0..n):seq(a(n)+sum(n*k, k=2..n), n=0..30); # Zerinvary Lajos, Jun 10 2008
a:=n->sum(-2+sum(2+sum(2, j=1..n),j=1..n),j=1..n):seq(a(n)/2,n=0..40);# Zerinvary Lajos, Dec 06 2008
seq(n^3+n^2-n, n=0..100); # Robert Israel, Dec 05 2014

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 33}, 60] (* Vincenzo Librandi, Jun 22 2017 *)

Formula

a(n) = n^3 + n^2 - n = n*A028387(n-1).

a(n) = A081437(n-1), n>0. - R. J. Mathar, Sep 12 2008

G.f.: x*(1+6*x-x^2)/(1-x)^4. - Robert Israel, Dec 05 2014

E.g.f.: x*(1+4*x+x^2)*exp(x). - Robert Israel, Dec 05 2014

For q a prime power, a(q) is the number of pairs of commuting nilpotent 2*2 matrices with coefficients in GL(q). (Proof: the zero matrix commutes with all q^2 nilpotent matrices, each of the remaining q^2-1 nilpotent matrices commutes with exactly q nilpotent matrices.) - Mark Wildon, Jun 18 2017