A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n*(n+1)*(10*n-7)/6.

0, 1, 13, 46, 110, 215, 371, 588, 876, 1245, 1705, 2266, 2938, 3731, 4655, 5720, 6936, 8313, 9861, 11590, 13510, 15631, 17963, 20516, 23300, 26325, 29601, 33138, 36946, 41035, 45415, 50096, 55088, 60401, 66045, 72030, 78366, 85063, 92131, 99580, 107420, 115661

Offset: 0

Author

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

Keywords

• nonn
• easy
• nice

Comments

Binomial transform of [1, 12, 21, 10, 0, 0, 0, ...] = (1, 13, 46, 110, ...). - Gary W. Adamson, Nov 28 2007

This sequence is related to A000566 by a(n) = n*A000566(n) - Sum_{i=0..n-1} A000566(i) and this is the case d=5 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/6. - Bruno Berselli, Oct 18 2010

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
• B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A093645 ((10, 1) Pascal, column m=3). Partial sums of A051624.

Cf. A000566.

See similar sequences listed in A237616.

Programs

GAP

List([0..45], n-> n*(n+1)*(10*n-7)/6); # G. C. Greubel, Aug 30 2019

Magma

[ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..45] ]; // Klaus Brockhaus, Nov 20 2008

Maple

A007587:=n->n*(n+1)*(10*n-7)/6: seq(A007587(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014

Mathematica

CoefficientList[Series[x(1+9x)/(1-x)^4, {x,0,45}], x] (* Vincenzo Librandi, Jun 20 2013 *)
Table[n(n+1)(10n-7)/6,{n,0,50}] (* Harvey P. Dale, Nov 13 2013 *)

PARI

a(n)=if(n,([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;1;13;46])[1,1],0) \\ Charles R Greathouse IV, Oct 07 2015

PARI

vector(45, n, n*(n-1)*(10*n-17)/6) \\ G. C. Greubel, Aug 30 2019

Sage

[n*(n+1)*(10*n-7)/6 for n in (0..45)] # G. C. Greubel, Aug 30 2019

Formula

a(n) = (10*n-7)*binomial(n+1, 2)/3.

G.f.: x*(1+9*x)/(1-x)^4.

a(n) = Sum_{k=0..n} k*(5*k-4). - Klaus Brockhaus, Nov 20 2008

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 23 2014

E.g.f.: exp(x)*x*(6 + 33*x + 10*x^2)/6. - Elmo R. Oliveira, Aug 04 2025