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) = 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
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Feb 11 2014
STATUS
approved