A237618 a(n) = n*(n + 1)*(19*n - 16)/6.

0, 1, 22, 82, 200, 395, 686, 1092, 1632, 2325, 3190, 4246, 5512, 7007, 8750, 10760, 13056, 15657, 18582, 21850, 25480, 29491, 33902, 38732, 44000, 49725, 55926, 62622, 69832, 77575, 85870, 94736, 104192, 114257, 124950, 136290, 148296, 160987, 174382

Offset: 0

Author

Bruno Berselli, Feb 11 2014

Keywords

• nonn
• easy

Comments

Also 21-gonal (or icosihenagonal) pyramidal numbers.

Examples

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  20;
   3,  40,  39;
   4,  60,  78,  58;
   5,  80, 117, 116, 77;
   6, 100, 156, 174, 154, 96;
   7, 120, 195, 232, 231, 192, 115;
   8, 140, 234, 290, 308, 288, 230, 134;
   9, 160, 273, 348, 385, 384, 345, 268, 153;
  10, 180, 312, 406, 462, 480, 460, 402, 306, 172; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 19*r-18 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.
		

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (nineteenth row of the table).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pyramidal Number.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A051873, A226490.

Cf. similar sequences listed in A237616.

Programs

Magma

[n*(n+1)*(19*n-16)/6: n in [0..40]];

Magma

I:=[0,1,22,82]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014

Mathematica

Table[n(n+1)(19n-16)/6, {n, 0, 40}]
CoefficientList[Series[x(1+18x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)

SageMath

b=binomial; [b(n+2,3) +18*b(n+1,3) for n in (0..50)] # G. C. Greubel, May 27 2022

Formula

G.f.: x*(1 + 18*x) / (1 - x)^4.

a(n) = (1/2)*( n*A226490(n) - Sum_{j=0..n-1} A226490(j) ).

a(n) = Sum_{i=0..n-1} (n-i)*(19*i+1), for n>0; see the generalization in A237616 (Formula field).

From G. C. Greubel, May 27 2022: (Start)

a(n) = binomial(n+2, 3) + 18*binomial(n+1, 3).

E.g.f.: (1/6)*x*(6 + 60*x + 19*x^2)*exp(x). (End)