A268685 a(n) = 3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4.

6, 126, 630, 1950, 4680, 9576, 17556, 29700, 47250, 71610, 104346, 147186, 202020, 270900, 356040, 459816, 584766, 733590, 909150, 1114470, 1352736, 1627296, 1941660, 2299500, 2704650, 3161106, 3673026, 4244730, 4880700, 5585580, 6364176, 7221456, 8162550

Offset: 0

Author

Ilya Gutkovskiy, Feb 11 2016

Keywords

• nonn
• easy

Comments

a(n) is the total volume of the family of (n+1) rectangular prisms, where the k-th prism has dimensions (3k) X (3k-1) X (3k-2). - Wesley Ivan Hurt, Oct 02 2018

Examples

			a(0) = 1*2*3 = 6;
a(1) = 1*2*3 + 4*5*6 = 126;
a(2) = 1*2*3 + 4*5*6 + 7*8*9 = 630;
a(3) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 = 1950;
a(4) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 = 4680;
a(5) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 = 9576, etc.
		

Links

• Felix Fröhlich, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000027, A054776, A116689, A135036.

Trisection of A319014 and A319867.

Programs

Magma

[3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4: n in [0..40]]; // Vincenzo Librandi, Feb 11 2016

Mathematica

Table[3 (n + 1) (n + 2) (3 n + 1) ((3 n + 4)/4), {n, 0, 32}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {6, 126, 630, 1950, 4680}, 32]
CoefficientList[Series[6 (10 x^2 + 16 x + 1) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 11 2016 *)

PARI

a(n) = 3*(n+1)*(n+2)*(3*n+1)*(3*n+4)/4 \\ Felix Fröhlich, Jun 07 2016

Formula

G.f.: -6*(10*x^2 + 16*x + 1)/(x - 1)^5.

a(n) = Sum_{k = 0..n} (3*k + 1)(3*k + 2)(3*k + 3).

Sum {n>=0} 1/a(n) = 2*(sqrt(3)*Pi + 9*log(3) - 14)/15 = 0.1771878254287521...

a(n) mod 6 = 0.

a(n) = 6*A116689(n+1). - R. J. Mathar, Jun 07 2016

E.g.f.: 3*exp(x)*(8 + 160*x +256*x^2 + 96*x^3 + 9*x^4)/4. - Stefano Spezia, Apr 18 2023

Sum_{n>=0} (-1)^n/a(n) = 28/15 - 8*Pi/(15*sqrt(3)) - 16*log(2)/15. - Amiram Eldar, Apr 30 2023