OFFSET
0,2
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).
FORMULA
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
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
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 *)
PROG
(PARI) a(n)=n^2*(n^2+1) \\ Charles R Greathouse IV, Sep 24 2015
(Magma)
A071253:= func< n | 2*Binomial(n^2+1, 2) >;
[A071253(n): n in [0..40]]; // G. C. Greubel, Sep 12 2024
(SageMath)
def A071253(n): return 2*binomial(n^2+1, 2)
[A071253(n) for n in range(41)] # G. C. Greubel, Sep 12 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 12 2002
STATUS
approved