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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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