A002414 Octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2. (Formerly M4609 N1966)

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

1,2

Comments

Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes. - Yong Kong (ykong@curagen.com), 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

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).

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

Example

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 (ykong@curagen.com), 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 + ...

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 *)

Prog

(PARI) {a(n) = (2*n - 1) * n * (n + 1) / 2} \\ Michael Somos, Mar 17 2011
(Magma) [n*(n+1)*(2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, May 24 2016

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)).

Sequence in context: A005919 A084370 A000439 * A273604 A273640 A344522

Adjacent sequences: A002411 A002412 A002413 * A002415 A002416 A002417

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

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

Incorrect formula deleted by Ant King, Oct 04 2012

Status

approved