A249127 a(n) = n * floor(3*n/2).

0, 1, 6, 12, 24, 35, 54, 70, 96, 117, 150, 176, 216, 247, 294, 330, 384, 425, 486, 532, 600, 651, 726, 782, 864, 925, 1014, 1080, 1176, 1247, 1350, 1426, 1536, 1617, 1734, 1820, 1944, 2035, 2166, 2262, 2400, 2501, 2646, 2752, 2904, 3015, 3174, 3290, 3456, 3577, 3750, 3876, 4056, 4187, 4374, 4510

Offset: 0

Author

Karl V. Keller, Jr., Oct 21 2014

Keywords

• nonn
• easy

Comments

Union of A033570, that is (2*n+1)*(3*n+1), and A033581, that is 6*n^2.

Examples

			For n=5, a(n) = 5*floor(15/2) = 5*7 = 35.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pentagonal Number
• Wikipedia, Pentagonal number
• Wikipedia, Platonic Solid
• Wolfram MathWorld, Platonic Solid
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A032766, A033581 (6*n^2), A033570 (2*n+1)*(3*n+1), A001318 (n*(3*n-1)/2).

Programs

Magma

[n*Floor(3*n/2): n in [0..60]]; // Vincenzo Librandi, Oct 22 2014

Maple

seq(n*floor(3*n/2), n=0..100); # Robert Israel, Oct 26 2014

Mathematica

Table[n Floor[3 n/2], {n, 0, 100}] (* Vincenzo Librandi, Oct 22 2014 *)

PARI

a(n)=3*n\2*n \\ Charles R Greathouse IV, Oct 21 2014

PARI

concat(0, Vec(-x*(2*x^3+4*x^2+5*x+1)/((x-1)^3*(x+1)^2) + O(x^100))) \\ Colin Barker, Oct 22 2014

Python

from math import *
{print(int(n*floor(3*n/2)),end=', ') for n in range(101)}

Formula

a(n) = n * floor(3n/2) = n * A032766(n).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Colin Barker, Oct 22 2014

G.f.: -x*(2*x^3+4*x^2+5*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Oct 22 2014

a(n) = 3/2 * n^2 + ((-1)^n-1) * n/4. E.g.f.: ((3/2)*x^2+(5/4)*x)*exp(x)-(x/4)*exp(-x). - Robert Israel, Oct 26 2014