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

0,3

Comments

Also 21-gonal (or icosihenagonal) pyramidal numbers.

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).

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)

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A051873, A226490.

Cf. similar sequences listed in A237616.

Sequence in context: A094844 A323251 A010010 * A105101 A235766 A044209

Adjacent sequences: A237615 A237616 A237617 * A237619 A237620 A237621

Keyword

nonn,easy

Author

Bruno Berselli, Feb 11 2014

Status

approved