A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n*(n+1)*(10*n-7)/6. (Formerly M4895)

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

0,3

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.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), pages 92-93.

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).

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

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 *)

Prog

(Magma) [ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..45] ]; // Klaus Brockhaus, Nov 20 2008
(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
(SageMath) [n*(n+1)*(10*n-7)/6 for n in (0..45)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..45], n-> n*(n+1)*(10*n-7)/6); # G. C. Greubel, Aug 30 2019

Crossrefs

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

Cf. A000566.

See similar sequences listed in A237616.

Sequence in context: A281315 A361622 A010003 * A318035 A219905 A326163

Adjacent sequences: A007584 A007585 A007586 * A007588 A007589 A007590

Keyword

nonn,easy,nice

Author

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

Status

approved