A002898 Number of n-step closed paths on hexagonal lattice.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• nice

Comments

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

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

Examples

			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
		

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
• Cyril Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001 (>6 Mb).
• Volker Braun, Philip Candelas, and Xenia de la Ossa, Two One-Parameter Special Geometries, arXiv preprint arXiv:1512.08367 [hep-th], 2015.
• Gunther Cornelissen, David Hokken, and Berend Ringeling, The asymptotic Mahler measure of Gaussian periods, arXiv:2507.09303 [math.NT], 2025. See p. 40.
• Cyril Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
• Davidson Noby Joseph and Igor Boettcher, Walking on Archimedean Lattices: Insights from Bloch Band Theory, arXiv:2507.12662 [cond-mat.stat-mech], 2025. See p. 18.
• 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, Reykjavík, 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.
• Gabriele 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.

Crossrefs

Cf. A000172, A006480, A337905-A337907, A094060, A002894 (returns square lattice), A002893 (honeycomb net).

Programs

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 *)

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 */

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

Extensions

More terms from David Bloom, Mar 1997

Formula and further terms from Cyril Banderier, Oct 12 2000