A007586 11-gonal (or hendecagonal) pyramidal numbers: n*(n+1)*(3*n-2)/2. (Formerly M4835)

0, 1, 12, 42, 100, 195, 336, 532, 792, 1125, 1540, 2046, 2652, 3367, 4200, 5160, 6256, 7497, 8892, 10450, 12180, 14091, 16192, 18492, 21000, 23725, 26676, 29862, 33292, 36975, 40920, 45136, 49632, 54417, 59500, 64890, 70596, 76627, 82992, 89700, 96760, 104181

Offset

0,3

Comments

Starting with 1 equals binomial transform of [1, 11, 19, 9, 0, 0, 0, ...]. - Gary W. Adamson, Nov 02 2007

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

G.f.: x*(1+8*x)/(1-x)^4.

a(0)=0, a(1)=1, a(2)=12, a(3)=42; for n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Apr 09 2012

a(n) = Sum_{i=0..n-1} (n-i)*(9*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014

From Amiram Eldar, Jun 28 2020: (Start)

Sum_{n>=1} 1/a(n) = (9*log(3) + sqrt(3)*Pi - 4)/10.

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)*Pi + 2 - 4*log(2))/5. (End)

Example

From Vincenzo Librandi, Feb 12 2014: (Start)
After 0, the sequence is provided by the row sums of the triangle (see above, third formula):
  1;
  2, 10;
  3, 20, 19;
  4, 30, 38, 28;
  5, 40, 57, 56, 37;
  6, 50, 76, 84, 74, 46; etc. (End)

Maple

seq(n*(n+1)*(3*n-2)/2, n=0..45); # G. C. Greubel, Aug 30 2019

Mathematica

Table[n(n+1)(3n-2)/2, {n, 0, 45}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 12, 42}, 45] (* Harvey P. Dale, Apr 09 2012 *)
CoefficientList[Series[x(1+8x)/(1-x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Feb 12 2014 *)

Prog

(Magma) I:=[0, 1, 12, 42]; [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
(PARI) a(n)=n*(n+1)*(3*n-2)/2 \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n*(n+1)*(3*n-2)/2 for n in (0..45)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..45], n-> n*(n+1)*(3*n-2)/2); # G. C. Greubel, Aug 30 2019

Crossrefs

Cf. A051682.

Cf. A093644 ((9, 1) Pascal, column m=3).

Cf. similar sequences listed in A237616.

Sequence in context: A090554 A009948 A193068 * A228391 A334277 A122973

Adjacent sequences: A007583 A007584 A007585 * A007587 A007588 A007589

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, R. K. Guy

Extensions

More terms from Vincenzo Librandi, Feb 12 2014

Status

approved