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A002414 Octagonal pyramidal numbers: n*(n+1)*(2*n-1)/2.
(Formerly M4609 N1966)
31
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of ways of covering 2n x 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes.

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(A000326(i), i=0..n-1) and this is the case d=3 in the identity n*(n*(d*n-d+2)/2)-sum(k*(d*k-d+2)/2, k=0..n-1) = 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

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.

M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.

P. W. Kasteleyn, The statistics of dimers on a lattice, Physica, 27(1961), 1209-1225.

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).

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.

Index entries for sequences related to dominoes

Index to sequences with 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>=0} 1/a(n) = 2*(4*log(2)-1)/3 = 1.1817258148265...

(End)

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

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.

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);

nmax:=38; for n from 0 to nmax do fz(n):=product((1-(k+1)*z)^(1+3*k), k=0..n); c(n):= abs(coeff(fz(n), z, 1)); end do: a:=n-> c(n): seq(a(n), n=0..nmax); # Johannes W. Meijer, Mar 07 2009

a:=n-> add (j*(n+1)+n*(j-1), j=0..n): seq(a(n), n=1..40); # Zerinvary Lajos, Apr 18 2009

MATHEMATICA

f[n_]:=6*n+1; s1=s2=0; lst={}; Do[a=f[n]; s1+=a; s2+=s1; AppendTo[lst, s2], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 25 2009 *)

Table[Sum[(n^2 - i), {i, 0, n}], {n, 1, 40}] (* Zerinvary Lajos, Jul 11 2009 *)

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 */

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.

Sequence in context: A005919 A084370 A000439 * A212517 A000440 A161684

Adjacent sequences:  A002411 A002412 A002413 * A002415 A002416 A002417

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Additional comments from Yong Kong (ykong(AT)curagen.com), May 06 2000

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

Incorrect formula deleted by Ant King, Oct 04 2012

STATUS

approved

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Last modified November 22 20:13 EST 2014. Contains 249827 sequences.