A265645 a(n) = n^2 * floor(n/2).

0, 0, 4, 9, 32, 50, 108, 147, 256, 324, 500, 605, 864, 1014, 1372, 1575, 2048, 2312, 2916, 3249, 4000, 4410, 5324, 5819, 6912, 7500, 8788, 9477, 10976, 11774, 13500, 14415, 16384, 17424, 19652, 20825, 23328, 24642, 27436, 28899, 32000, 33620, 37044, 38829, 42592, 44550, 48668, 50807, 55296, 57624, 62500

Offset: 0

Author

Ian Stewart, Apr 06 2016

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A000290, A004526.

Programs

GAP

List([0..55], n -> n^2*Int(n/2)); # Muniru A Asiru, Apr 05 2018

Haskell

seq' x = x^2 * (x `div` 2)
map seq' [0..50]

Magma

[n^2*Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Apr 04 2018

Mathematica

Table[n^2 Floor[n/2], {n, 0, 50}] (* Vincenzo Librandi, Apr 04 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,0,4,9,32,50,108},60] (* Harvey P. Dale, May 19 2019 *)

PARI

a(n) = n^2*(n\2); \\ Altug Alkan, Apr 14 2016

PARI

concat(vector(2), Vec(x^2*(4+5*x+11*x^2+3*x^3+x^4)/((1-x)^4*(1+x)^3) + O(x^50))) \\ Colin Barker, Apr 14 2016

Formula

a(n) = A000290(n)*A004526(n). - Michel Marcus, Apr 14 2016

G.f.: x^2*(4 + 5*x + 11*x^2 + 3*x^3 + x^4)/((1 - x)^4*(1 + x)^3). - Ilya Gutkovskiy, Apr 14 2016; corrected by Colin Barker, Apr 14 2016

From Colin Barker, Apr 14 2016: (Start)

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

a(n) = n^3/2 for n even.

a(n) = n^2*(n-1)/2 for n odd.

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>6. (End)

Sum_{n>=2} 1/a(n) = zeta(3)/4 - Pi^2/4 - 2*log(2) + 4. - Amiram Eldar, Mar 15 2024