A264851 a(n) = n*(n + 1)*(n + 2)*(4*n - 3)/6.

0, 1, 20, 90, 260, 595, 1176, 2100, 3480, 5445, 8140, 11726, 16380, 22295, 29680, 38760, 49776, 62985, 78660, 97090, 118580, 143451, 172040, 204700, 241800, 283725, 330876, 383670, 442540, 507935, 580320, 660176, 748000, 844305, 949620, 1064490, 1189476, 1325155

Offset

0,3

Comments

Partial sums of 18-gonal (or octadecagonal) pyramidal numbers. Therefore, this is the case k=8 of the general formula n*(n + 1)*(n + 2)*(k*n - k + 2)/12, which is related to 2*(k+1)-gonal pyramidal numbers.

Links

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000

OEIS Wiki, Figurate numbers.

Eric Weisstein's World of Mathematics, Pyramidal Number.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: x*(1 + 15*x)/(1 - x)^5.

a(n) = Sum_{k = 0..n} A172078(k).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

From Amiram Eldar, Feb 20 2026: (Start)

Sum_{n>=1} 1/a(n) = (16*Pi + 96*log(2) - 69/2)/77.

Sum_{n>=1} (-1)^(n+1)/a(n) = (153 + 32*sqrt(2)*Pi + 8*(4*sqrt(2)-41)*log(2) - 64*sqrt(2)*log(2-sqrt(2)))/154. (End)

Mathematica

Table[n (n + 1) (n + 2) (4 n - 3)/6, {n, 0, 50}]

Prog

(Magma) [n*(n + 1)*(n + 2)*(4*n - 3)/6: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015
(PARI) a(n)=n*(n+1)*(n+2)*(4*n-3)/6 \\ Charles R Greathouse IV, Jul 26 2016

Crossrefs

Cf. A172078.

Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

Sequence in context: A264302 A220200 A242656 * A345286 A338485 A344334

Adjacent sequences: A264848 A264849 A264850 * A264852 A264853 A264854

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 26 2015

Status

approved