

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
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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(n1).
a(n) = A081437(n1), n>0.  R. J. Mathar, Sep 12 2008
G.f.: x*(1+6*xx^2)/(1x)^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^21 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^2n, 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^2n: 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



