A237617 a(n) = n*(n + 1)*(17*n - 14)/6.

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

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