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 A237617 a(n) = n*(n + 1)*(17*n - 14)/6. 2
 0, 1, 20, 74, 180, 355, 616, 980, 1464, 2085, 2860, 3806, 4940, 6279, 7840, 9640, 11696, 14025, 16644, 19570, 22820, 26411, 30360, 34684, 39400, 44525, 50076, 56070, 62524, 69455, 76880, 84816, 93280, 102289, 111860, 122010, 132756, 144115, 156104, 168740 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also 19-gonal (or nonadecagonal) pyramidal numbers. This sequence is related to A180232 by 2*a(n) = n*A180232(n) - Sum_{i=0..n-1} A180232(i). REFERENCES E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (seventeenth 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 to sequences related to pyramidal numbers Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA G.f.: x*(1 + 16*x)/(1 - x)^4. For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(17*i+1); see the generalization in A237616 (Formula field). E.g.f.: x*(6 + 54*x + 17*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019 EXAMPLE After 0, the sequence is provided by the row sums of the triangle: 1; 2, 18; 3, 36, 35; 4, 54, 70, 52; 5, 72, 105, 104, 69; 6, 90, 140, 156, 138, 86; 7, 108, 175, 208, 207, 172, 103; 8, 126, 210, 260, 276, 258, 206, 120; 9, 144, 245, 312, 345, 344, 309, 240, 137; 10, 162, 280, 364, 414, 430, 412, 360, 274, 154; etc., where (r = row index, c = column index): T(r,r) = T(c,c) = 17*r - 16 and T(r,c) = T(r-1,c) + T(r,r) = (r-c+1) * T(r,r), with r>=c>0. MAPLE seq(n*(n+1)*(17*n-14)/6, n=0..40); # G. C. Greubel, Aug 30 2019 MATHEMATICA Table[n(n+1)(17*n-14)/6, {n, 0, 40}] CoefficientList[Series[x(1+16x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 20, 74}, 40] (* Harvey P. Dale, Aug 04 2021 *) PROG (Magma) [n*(n+1)*(17*n-14)/6: n in [0..40]]; (Magma) I:=[0, 1, 20, 74]; [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..40]]; // Vincenzo Librandi, Feb 12 2014 (PARI) vector(40, n, n*(n-1)*(17*n-31)/6) \\ G. C. Greubel, Aug 30 2019 (Sage) [n*(n+1)*(17*n-14)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019 (GAP) List([0..40], n-> n*(n+1)*(17*n-14)/6); # G. C. Greubel, Aug 30 2019 CROSSREFS Cf. A051871, A180232. Cf. similar sequences listed in A237616. Sequence in context: A225923 A238026 A010008 * A000529 A238027 A005565 Adjacent sequences: A237614 A237615 A237616 * A237618 A237619 A237620 KEYWORD nonn,easy AUTHOR Bruno Berselli, Feb 11 2014 STATUS approved

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Last modified October 1 11:17 EDT 2023. Contains 365826 sequences. (Running on oeis4.)