A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6.

0, 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, 2380, 2920, 3536, 4233, 5016, 5890, 6860, 7931, 9108, 10396, 11800, 13325, 14976, 16758, 18676, 20735, 22940, 25296, 27808, 30481, 33320, 36330, 39516, 42883, 46436, 50180, 54120

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The partial sums of A000566. - R. J. Mathar, Mar 19 2008

A002413(n + 1) is the number of 4-tuples (w, x, y, z) having all terms in {0, ..., n} and w = floor((x + y + z)/2). - Clark Kimberling, May 28 2012

From Ant King, Oct 25 2012: (Start)

For n > 0, the digital roots of this sequence A010888(A002413(n)) form the purely periodic 27-cycle {1, 8, 8, 6, 7, 7, 2, 6, 6, 7, 5, 5, 3, 4, 4, 8, 3, 3, 4, 2, 2, 9, 1, 1, 5, 9, 9}.

For n > 0, the units' digits of this sequence A010879(A002413(n)) form the purely periodic 20-cycle {1, 8, 6, 0, 5, 6, 8, 6, 5, 0, 6, 8, 1, 0, 0, 6, 3, 6, 0, 0}.

(End)

Examples

			For n=7, a(7) = 7*1 + 6*6 + 5*11 + 4*16 + 3*21 + 2*26 + 1*31 = 308. - _Bruno Berselli_, Feb 10 2014
		

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.
• L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number.
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A093562 ((5, 1) Pascal, column m = 3).

Cf. similar sequences listed in A237616.

Programs

Magma

[n*(n + 1)*(5*n - 2)/6: n in [0..50]]; // G. C. Greubel, Nov 04 2017

Maple

A002413:=n->n*(n+1)*(5*n-2)/6: seq(A002413(n), n=0..60); # Wesley Ivan Hurt, Apr 14 2017

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 8, 26, 60}, 40] (* Ant King, Oct 25 2012 *)
Table[(5n^3 + 3n^2 - 2n)/6, {n, 0, 39}] (* Alonso del Arte, Oct 25 2012 *)

Maxima

A002413(n):=n*(n+1)*(5*n-2)/6$ makelist(A002413(n),n,0,20); /* Martin Ettl, Dec 12 2012 */

PARI

a(n)=n*(n+1)*(5*n-2)/6 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = n*(n + 1)*(5*n - 2)/6.

G.f.: x*(1 + 4*x)/(1 - x)^4. [Suggested by Simon Plouffe in his 1992 dissertation.]

From Ant King, Oct 25 2012: (Start)

a(n) = a(n - 1) + n*(5*n - 3)/2.

a(n) = 3*a(n - 1) - 3*a(n - 2) + a(n - 3) + 5.

a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4)

a(n) = (n + 1)*(2*A000566(n) + n)/6 = (5*n - 2)*A000217(n)/3.

a(n) = A000292(n) + 4*A000292(n - 1)

a(n) = A002412(n) + A000292(n - 1)

a(n) = A000217(n) + 5*A000292(n - 1)

a(n) = binomial(n + 2, 3) + 4*binomial(n + 1, 3) = (5*n - 2) * binomial(n + 1, 2)/3.

Sum_{n >= 1} 1/a(n) = 15*(log(3125) + sqrt(5)*log((3 - sqrt(5))/2) - 2*Pi*sqrt(5*(5 - 2*sqrt(5)))/5 - 8/5)/28 = 1.207293...

(End)

a(n) = Sum_{i=0..n-1} (n-i)*(5*i+1). - Bruno Berselli, Feb 10 2014

a(n) = A080851(5,n-1). - R. J. Mathar, Jul 28 2016

E.g.f.: x*(6 + 18*x + 5*x^2)*exp(x)/6. - Ilya Gutkovskiy, May 12 2017

a(n) = Sum_{i=0..n-1} (n+2*i)*(n-i). - Leonid Bedratyuk, Jul 09 2024

Extensions

More terms from James Sellers, Dec 23 1999

a(0)=0 prepended by Max Alekseyev, Nov 23 2011