A071253 a(n) = n^2*(n^2+1).

0, 2, 20, 90, 272, 650, 1332, 2450, 4160, 6642, 10100, 14762, 20880, 28730, 38612, 50850, 65792, 83810, 105300, 130682, 160400, 194922, 234740, 280370, 332352, 391250, 457652, 532170, 615440, 708122, 810900, 924482, 1049600, 1187010, 1337492, 1501850, 1680912

Offset: 0

Author

N. J. A. Sloane, Jun 12 2002

Keywords

• nonn
• easy

Comments

The identity (n^5 + n^3)^2 + (n^2*(n^2 + 1))^2 = n*(n^3 + n)^3 can be written as A155977(n)^2 + a(n)^2 = n*A034262(n)^3. - Vincenzo Librandi, Aug 08 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000290, A002522, A002378, A034262, A037270, A069187, A155977.

Programs

Magma

A071253:= func< n | 2*Binomial(n^2+1,2) >;
[A071253(n): n in [0..40]]; // G. C. Greubel, Sep 12 2024

Maple

with(combinat):seq(lcm(fibonacci(3,n),n^2),n=0..35); # Zerinvary Lajos, Apr 20 2008
a:=n->add(n+add(n+add(n, j=1..n-1),j=1..n),j=1..n):seq(a(n), n=0..41); # Zerinvary Lajos, Aug 27 2008

Mathematica

Table[(1/4) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n,0,40}] (* Artur Jasinski, Feb 10 2010 *)
Table[n^2*(n^2+1),{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[2 x (1+x) (1+4 x+x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

PARI

a(n)=n^2*(n^2+1) \\ Charles R Greathouse IV, Sep 24 2015

SageMath

def A071253(n): return 2*binomial(n^2+1,2)
[A071253(n) for n in range(41)] # G. C. Greubel, Sep 12 2024

Formula

a(n) = A002522(n)*A000290(n). - Zerinvary Lajos, Apr 20 2008

a(n) = (1/4)*sinh(2*arcsinh(n))^2. - Artur Jasinski, Feb 10 2010

G.f.: 2*x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Jan 08 2012

a(n) = A002378(A000290(n)). - Rick L. Shepherd, Sep 22 2014

Sum_{n>=1} 1/a(n) = 0.5682... = Pi^2/6- (Pi*coth Pi-1)/2 = A013661 - A259171 [J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Oct 18 2019

a(n) = 2*A037270(n). - R. J. Mathar, Oct 18 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 1/2 + Pi*cosech(Pi)/2. - Amiram Eldar, Nov 05 2020

E.g.f.: exp(x)*x*(2 + 8*x + 6*x^2 + x^3). - Stefano Spezia, Oct 08 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 16 2023