A002414 Octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2.

1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, 1386, 1794, 2275, 2835, 3480, 4216, 5049, 5985, 7030, 8190, 9471, 10879, 12420, 14100, 15925, 17901, 20034, 22330, 24795, 27435, 30256, 33264, 36465, 39865, 43470, 47286, 51319, 55575, 60060, 64780

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes. - Yong Kong ([e-mail]), May 06 2000

Equals binomial transform of [0, 1, 7, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jun 14 2008, corrected Oct 25 2012

Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF3 denominators of A156927. See A157704 for background information. - Johannes W. Meijer, Mar 07 2009

This sequence is related to A000326 by a(n) = n*A000326(n) - Sum_{i=0..n-1} A000326(i) and this is the case d=3 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, Apr 21 2010

2*a(n) gives the principal diagonal of the convolution array A213819. - Clark Kimberling, Jul 04 2012

Partial sums of the figurate octagonal numbers A000567. For each sequence with a linear recurrence with constant coefficients, the values reduced modulo some constant m generate a periodic sequence. For this sequence, these Pisano periods have length 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, ... for m >= 1. - Ant King, Oct 26 2012

Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016

On a square grid of side length n+1, the number of embedded rectangles (where each side is greater than 1). For example, in a 2 X 2 square there is one rectangle, in a 3 X 3 square there are nine rectangles, etc. - Peter Woodward, Nov 26 2017

a(n) is the sum of the numbers in the n X n square array A204154(n). - Ali Sada, Jun 21 2019

Sum of all multiples of n and n+1 that are <= n^2. - Wesley Ivan Hurt, May 25 2023

Examples

			a(2) = 9 since there are 9 ways to cover a 4 X 4 lattice with 8 dominoes, 2 of which is horizontal and the other 6 are vertical. - Yong Kong ([e-mail]), May 06 2000
G.f. = x + 9*x^2 + 30*x^3 + 70*x^4 + 135*x^5 + 231*x^6 + 364*x^7 + 540*x^8 + 765*x^9 + ...
		

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 = 1..1000
• B. Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
• M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
• P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
• 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
• Index entries for sequences related to dominoes
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000578, A004003, A160378.

Cf. A093563 (( 6, 1) Pascal, column m=3). A000567 (differences).

Cf. A156927, A157704. - Johannes W. Meijer, Mar 07 2009

Cf. A000326.

Cf. similar sequences listed in A237616.

Cf. A260234 (largest prime factor of a(n+1)).

Programs

Magma

[n*(n+1)*(2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, May 24 2016

Maple

A002414 := n-> 1/2*n*(n+1)*(2*n-1): seq(A002414(n), n=1..100);

Mathematica

LinearRecurrence[{4,-6,4,-1},{1,9,30,70},40] (* Harvey P. Dale, Apr 12 2013 *)

PARI

{a(n) = (2*n - 1) * n * (n + 1) / 2} \\ Michael Somos, Mar 17 2011

Formula

a(n) = odd numbers * triangular numbers = (2*n-1)* binomial(n+1,2). - Xavier Acloque, Oct 27 2003

G.f.: x*(1+5*x)/(1-x)^4. [Conjectured by Simon Plouffe in his 1992 dissertation.]

a(n) = A000578(n) + A000217(n-1). - Kieren MacMillan, Mar 19 2007

a(-n) = -A160378(n). - Michael Somos, Mar 17 2011

From Ant King, Oct 26 2012: (Start)

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

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

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

a(n) = n*A000326(n) - A002411(n-1), see Berselli's comment.

a(n) = (n+1)*(2*A000567(n)+n)/6.

a(n) = A000292(n) + 5*A000292(n-1) = binomial(n+2,3)+5*binomial(n+1,3).

a(n) = A002413(n) + A000292(n-1).

a(n) = A000217(n) + 6*A000292(n-1).

Sum_{n>=1} 1/a(n) = 2*(4*log(2)-1)/3 = 1.1817258148265...

(End)

a(n) = Sum_{i=0..n-1} (n-i)*(6*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014

E.g.f.: x*(2 + 7*x + 2*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 23 2016

a(n) = A080851(6,n-1). - R. J. Mathar, Jul 28 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi + 1 - 4*log(2))/3. - Amiram Eldar, Jul 02 2020

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000

Incorrect formula deleted by Ant King, Oct 04 2012