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A002413 Heptagonal (or 7-gonal) pyramidal numbers: n*(n+1)*(5*n-2)/6.
(Formerly M4498 N1904)
21
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 (list; graph; refs; listen; history; text; internal format)
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 A01088(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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number.

Index to sequences with 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*(5n - 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 >= 0} 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( (n-i)*(5*i+1), i=0..n-1 ), with a(0)=0. [Bruno Berselli, Feb 10 2014]

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]

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 */

CROSSREFS

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

Cf. similar sequences listed in A237616.

Sequence in context: A215097 A111694 A129111 * A218325 A163121 A250352

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

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Last modified November 26 10:12 EST 2014. Contains 250038 sequences.