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A268685
a(n) = 3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/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
0,1
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
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
EXAMPLE
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.
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 *)
PROG
(Magma) [3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4: n in [0..40]]; // 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
CROSSREFS
Trisection of A319014 and A319867.
Sequence in context: A224089 A331837 A254544 * A109820 A228290 A370715
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Feb 11 2016
STATUS
approved