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A002898 Number of n-step closed paths on hexagonal lattice.
(Formerly M4101 N1701)
22
1, 0, 6, 12, 90, 360, 2040, 10080, 54810, 290640, 1588356, 8676360, 47977776, 266378112, 1488801600, 8355739392, 47104393050, 266482019232, 1512589408044, 8610448069080, 49144928795820, 281164160225520, 1612061452900080, 9261029179733760, 53299490722049520 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also, number of closed paths of length n on the honeycomb lattice.

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

From David Callan, Aug 25 2009: (Start)

a(n) = number of 2 X n matrices, entries from {1,2,3}, second row a (multiset) permutation of the first, with no constant columns. For example, a(2)=6 counts the matrices

   1 2   1 3   2 1   2 3   3 1   3 2

   2 1   3 1   1 2   3 2   1 3   2 3. (End)

Also, a(n) is the constant coefficient in the expansion of (x + 1/x + y + 1/y + x/y + y/x)^n. - Christopher J. Smyth, Sep 25 2018

a(n) is the constant term in the expansion of (-2 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019

a(n) is the number of paths from (0,0,0) to (n,n,n) using the six permutations of (0,1,2) as steps, i.e., the steps (0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), and (2,1,0). - William J. Wang, Dec 07 2020

REFERENCES

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001 (>6 Mb).

V. Braun, P. Candelas, and X. de la Ossa, Two One-Parameter Special Geometries, arXiv preprint arXiv:1512.08367 [hep-th], 2015.

C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.

Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, Solution to 1995 Putnam problem A-6, Am. Math. Monthly, 1996, p. 674.

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjav´k, Iceland DMTCS proc. AO, 2011, 599-610.

Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Grzegorz Siudem and Agata Fronczak, Bell polynomials in the series expansions of the Ising model, arXiv:2007.16132 [math-ph], 2020.

FORMULA

D-finite with recurrence a(0) = 1, a(1) = 0, a(2) = 6, 36*(n+2)*(n+1)*a(n) +24*(n+2)^2*a(n+1) +(n+3)*(n+2)*a(n+2) -(n+3)^2*a(n+3) = 0.

E.g.f.: (BesselI(0,2*x))^3 + 2*Sum_{k>=1} (BesselI(k,2*x))^3. - Karol A. Penson Aug 18 2006

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n, i)*(Sum_{j=0..i} binomial(i, j)^3). - Vasu Tewari (vasu(AT)math.ubc.ca), Aug 04 2010

O.g.f.: (4/Pi)*EllipticK( 8*sqrt(z^3*(1+3*z))/(1-12*z^2+sqrt((1-6*z)*(1+2*z)^3)) ) / sqrt(2 - 24*z^2 + 2*sqrt((1-6*z)*(1+2*z)^3)). - Sergey Perepechko, Feb 08 2011

O.g.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+2*x)^n. - Paul D. Hanna, Feb 26 2012

a(n) ~ sqrt(3)*6^n/(2*Pi*n). - Vaclav Kotesovec, Aug 13 2013

O.g.f.: 2F1(1/3,2/3; 1; 27*x^2*(1+2*x)). - R. J. Mathar, Sep 29 2020

EXAMPLE

O.g.f.: 1 + 6*x^2 + 12*x^3 + 90*x^4 + 360*x^5 + 2040*x^6 + ...

O.g.f.: 1 + 6*x^2*(1+2*x) + 90*x^4*(1+2*x)^2 + 1680*x^6*(1+2*x)^3 + 34650*x^8*(1+2*x)^4 + ... + A006480(n)*x^(2*n)*(1+2*x)^n + .... - Paul D. Hanna, Feb 26 2012

MAPLE

a:= proc(n) option remember; `if`(n<3, [1, 0, 6][n+1], ((n-1)*

      n*a(n-1) +24*(n-1)^2*a(n-2) +36*(n-1)*(n-2)*a(n-3))/n^2)

    end:

seq(a(n), n=0..25);  # Alois P. Heinz, Dec 08 2020

MATHEMATICA

a[n_] := Sum[(-2)^(n-i)*Binomial[i, j]^3*Binomial[n, i], {i, 0, n}, {j, 0, i}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 21 2011, after Vasu Tewari *)

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, (3*m)!/m!^3*x^(2*m)*(1+2*x+x*O(x^n))^m), n)} /* Paul D. Hanna, Feb 26 2012 */

CROSSREFS

Cf. A000172, A006480, A337905-A337907, A094060, A002894 (returns square lattice).

Sequence in context: A005402 A128953 A181597 * A003613 A099767 A191462

Adjacent sequences:  A002895 A002896 A002897 * A002899 A002900 A002901

KEYWORD

nonn,walk,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David Bloom, Mar 1997

Formula and further terms from Cyril Banderier, Oct 12 2000

STATUS

approved

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Last modified February 28 14:02 EST 2021. Contains 341707 sequences. (Running on oeis4.)