A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6. (Formerly M4498 N1904)

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

0,3

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)

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).

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

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. similar sequences listed in A237616.

Sequence in context: A331242 A111694 A129111 * A218325 A363288 A252870

Adjacent sequences: A002410 A002411 A002412 * A002414 A002415 A002416

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Dec 23 1999

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

Status

approved