A181900 a(n) = n * A022998(n).

0, 1, 8, 9, 32, 25, 72, 49, 128, 81, 200, 121, 288, 169, 392, 225, 512, 289, 648, 361, 800, 441, 968, 529, 1152, 625, 1352, 729, 1568, 841, 1800, 961, 2048, 1089, 2312, 1225, 2592, 1369, 2888, 1521, 3200, 1681, 3528, 1849, 3872, 2025, 4232, 2209, 4608, 2401

Offset

0,3

Comments

Multiplicative because A022998 is. - Andrew Howroyd, Jul 26 2018

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

a(2*n) = 8*n^2 = A139098(n).

a(2*n+1) = 1 + 4*n + 4*n^2 = A016754(n).

From Colin Barker, Apr 01 2012: (Start)

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

G.f.: x*(1+8*x+6*x^2+8*x^3+x^4)/((1-x)^3*(1+x)^3). (End)

a(n) = numerator(2*n^2/(n^2+1)). - Vincenzo Librandi, Aug 19 2014

From Amiram Eldar, Feb 22 2022: (Start)

Sum_{n>=1} 1/a(n) = 7*Pi^2/48.

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/48. (End)

From G. C. Greubel, Aug 01 2022: (Start)

a(n) = n^2*(3 + (-1)^n)/2.

E.g.f.: (1/2)*x*(-1 + x + 3*(1 + x)*exp(2*x)). (End)

Mathematica

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 8, 9, 32, 25}, 50] (* Harvey P. Dale, Dec 01 2018 *)

Prog

(Haskell)
a181900 n = a022998 n * n
(PARI) a(n)=if(n%2, n^2, 2*n^2) \\ Charles R Greathouse IV, Aug 07 2012
(Magma) [Numerator(2*n^2/(n^2+1)): n in [0..50]]; // Vincenzo Librandi, Aug 19 2014
(SageMath) [n^2*(1 + ((n+1)%2)) for n in (0..60)] # G. C. Greubel, Aug 01 2022

Crossrefs

Cf. A016754, A022998, A086500 (partial sums), A139098.

Sequence in context: A303262 A225364 A304922 * A371443 A038344 A306162

Adjacent sequences: A181897 A181898 A181899 * A181901 A181902 A181903

Keyword

nonn,easy,mult

Author

Reinhard Zumkeller, Mar 31 2012

Status

approved