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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

Cf. A270109.

Sequence in context: A299287 A299285 A081437 * A162433 A003012 A020478

Adjacent sequences:  A085487 A085488 A085489 * A085491 A085492 A085493

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified December 5 21:00 EST 2019. Contains 329779 sequences. (Running on oeis4.)