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A237618 a(n) = n*(n + 1)*(19*n - 16)/6. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also 21-gonal (or icosihenagonal) pyramidal numbers.

This sequence is related to A226490 by 2*a(n) = n*A226490(n) - sum(A226490(i), i=0..n-1).

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.

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

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) (19 n - 16)/6, {n, 0, 40}]

CoefficientList[Series[x (1 + 18 x)/(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

CROSSREFS

Cf. A051873, A226490.

Cf. similar sequences listed in A237616.

Sequence in context: A253304 A094844 A010010 * A105101 A235766 A044209

Adjacent sequences:  A237615 A237616 A237617 * A237619 A237620 A237621

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Feb 11 2014

STATUS

approved

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Last modified June 24 08:33 EDT 2017. Contains 288697 sequences.