A082106 Main diagonal of number array A082105.

1, 6, 33, 118, 321, 726, 1441, 2598, 4353, 6886, 10401, 15126, 21313, 29238, 39201, 51526, 66561, 84678, 106273, 131766, 161601, 196246, 236193, 281958, 334081, 393126, 459681, 534358, 617793, 710646, 813601, 927366, 1052673, 1190278, 1340961, 1505526, 1684801

Offset: 0

Author

Paul Barry, Apr 03 2003

Keywords

• nonn
• easy

Comments

4*a(n) can be written as (n^2 + 2*n + 1)^2 + (n^2 - 2*n + 1)^2 + (n^2 - 2*n - 1)^2 + (n^2 + 2*n - 1)^2. - Bruno Berselli, Jun 20 2014

Links

• G. C. Greubel, 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. A082047, A082044, A082105, A082107, A082109.

Programs

Magma

[(n^2+2)^2 -3: n in [0..40]]; // G. C. Greubel, Dec 22 2022

Mathematica

Table[n^4+4n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,6,33,118,321},40] (* Harvey P. Dale, Dec 06 2012 *)

SageMath

[(n^2+2)^2 -3 for n in range(41)] # G. C. Greubel, Dec 22 2022

Formula

a(n) = n^4 + 4*n^2 + 1.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Dec 06 2012

G.f.: (1 + x + 13*x^2 + 3*x^3 + 6*x^4)/(1 - x)^5. - Bruno Berselli, Jun 20 2014

E.g.f.: (1 + 5*x + 11*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 22 2022

Sum_{n>=0} 1/a(n) = 1/2 + (Pi/4)*((1/sqrt(2)+1/sqrt(6))*coth(sqrt(2-sqrt(3))*Pi) - (1/sqrt(2)-1/sqrt(6))*coth(sqrt(2+sqrt(3))*Pi)). - Amiram Eldar, Jan 08 2023