A059826 a(n) = (n^2 - n + 1)*(n^2 + n + 1).

1, 3, 21, 91, 273, 651, 1333, 2451, 4161, 6643, 10101, 14763, 20881, 28731, 38613, 50851, 65793, 83811, 105301, 130683, 160401, 194923, 234741, 280371, 332353, 391251, 457653, 532171, 615441, 708123, 810901, 924483, 1049601, 1187011, 1337493, 1501851, 1680913

Offset: 0

Author

N. J. A. Sloane, Feb 24 2001

Keywords

• nonn
• easy

Comments

Main diagonal of A082039. - Paul Barry, Apr 02 2003

The base of the natural logarithms e = 2*Sum_{n>=0} 1/(a(n)*n!) and zeta(2) = Pi^2/6 = 1 + 2*Sum_{n>=1} (-1)^(n+1)/(a(n)*n^2). - Peter Bala, Jan 20 2008

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A002061, A062159.

Main diagonal of A082039.

Programs

Magma

[n^4+n^2+1 : n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

Maple

with(combinat): seq(fibonacci(3,n)+n^4, n=0..40); # Zerinvary Lajos, May 25 2008

Mathematica

Table[n^4 + n^2 + 1, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 09 2014 *)

PARI

a(n) = { my(f=n^2 + 1); (f - n)*(f + n) } \\ Harry J. Smith, Jun 29 2009

Formula

a(n) = n^4+n^2+1. - Paul Barry, Apr 02 2003

a(n) = (n^2-n+1) * (n^2+n+1) = A002061(n) * A002061(n+1), products of two consecutive central polygonal numbers. a(n) = (n^6-1)/(n^2-1), n>1. a(n) = (n^5-n^4+n^3-n^2+n-1)/(n-1) = A062159(n)/(n-1), n>1. - Alexander Adamchuk, Apr 12 2006

O.g.f.: (-1+2*x-16*x^2-6*x^3-3*x^4) / (x-1)^5. - R. J. Mathar, Feb 26 2008

a(n) = A219069(n,1), for n > 0. - Reinhard Zumkeller, Nov 11 2012

a(n+2) = (n^2+3n+3) * (n^2+5n+7) = (t(n)+t(n+2)) * (t(n+1)+t(n+3)), where t=A000217 are triangular numbers. For n>=1, a(n+2) = t(2*t(n+2)+t(n)) -t(t(n)-1). - J. M. Bergot, Nov 29 2012

4*a(n) = (n^2+n+1)^2+(n^2-n+1)^2+(n^2+n-1)^2+(n^2-n-1)^2. - Bruno Berselli, Jul 03 2014

a(n) = A002061(n^2). - Franklin T. Adams-Watters, Aug 01 2014

Sum_{n>=0} 1/a(n) = 1/2 + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6. - Amiram Eldar, Feb 14 2021