cp's OEIS Frontend

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

%I A002898 M4101 N1701 #124 Jul 23 2025 10:03:56
%S A002898 1,0,6,12,90,360,2040,10080,54810,290640,1588356,8676360,47977776,
%T A002898 266378112,1488801600,8355739392,47104393050,266482019232,
%U A002898 1512589408044,8610448069080,49144928795820,281164160225520,1612061452900080,9261029179733760,53299490722049520
%N A002898 Number of n-step closed paths on hexagonal lattice.
%C A002898 Also, number of closed paths of length n on the honeycomb tiling.
%C A002898 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C A002898 From _David Callan_, Aug 25 2009: (Start)
%C A002898 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
%C A002898    1 2   1 3   2 1   2 3   3 1   3 2
%C A002898    2 1   3 1   1 2   3 2   1 3   2 3. (End)
%C A002898 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
%C A002898 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
%C A002898 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
%D A002898 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002898 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002898 Vincenzo Librandi, <a href="/A002898/b002898.txt">Table of n, a(n) for n = 0..200</a>
%H A002898 Cyril Banderier, <a href="http://lipn.univ-paris13.fr/~banderier/Papers/these.ps">Analytic combinatorics of random walks and planar maps</a>, PhD Thesis, 2001 (>6 Mb).
%H A002898 Volker Braun, Philip Candelas, and Xenia de la Ossa, <a href="https://arxiv.org/abs/1512.08367">Two One-Parameter Special Geometries</a>, arXiv preprint arXiv:1512.08367 [hep-th], 2015.
%H A002898 Gunther Cornelissen, David Hokken, and Berend Ringeling, <a href="https://arxiv.org/abs/2507.09303">The asymptotic Mahler measure of Gaussian periods</a>, arXiv:2507.09303 [math.NT], 2025. See p. 40.
%H A002898 Cyril Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., 9 (1960), 149-361.
%H A002898 Davidson Noby Joseph and Igor Boettcher, <a href="https://arxiv.org/abs/2507.12662">Walking on Archimedean Lattices: Insights from Bloch Band Theory</a>, arXiv:2507.12662 [cond-mat.stat-mech], 2025. See p. 18.
%H A002898 Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, <a href="http://www.jstor.org/stable/2974878">Solution to 1995 Putnam problem A-6</a>, Am. Math. Monthly, 1996, p. 674.
%H A002898 Gilbert Labelle and Annie Lacasse, <a href="https://doi.org/10.46298/dmtcs.2937">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavík, Iceland DMTCS proc. AO, 2011, 599-610.
%H A002898 Yen Lee Loh, <a href="https://arxiv.org/abs/1706.03083">A general method for calculating lattice Green functions on the branch cut</a>, arXiv:1706.03083 [math-ph], 2017.
%H A002898 Gabriele Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H A002898 Grzegorz Siudem and Agata Fronczak, <a href="https://arxiv.org/abs/2007.16132">Bell polynomials in the series expansions of the Ising model</a>, arXiv:2007.16132 [math-ph], 2020.
%F A002898 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.
%F A002898 E.g.f.: (BesselI(0,2*x))^3 + 2*Sum_{k>=1} (BesselI(k,2*x))^3. - _Karol A. Penson_ Aug 18 2006
%F A002898 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
%F A002898 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
%F A002898 O.g.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+2*x)^n. - _Paul D. Hanna_, Feb 26 2012
%F A002898 a(n) ~ sqrt(3)*6^n/(2*Pi*n). - _Vaclav Kotesovec_, Aug 13 2013
%F A002898 O.g.f.: 2F1(1/3,2/3; 1; 27*x^2*(1+2*x)). - _R. J. Mathar_, Sep 29 2020
%e A002898 O.g.f.: 1 + 6*x^2 + 12*x^3 + 90*x^4 + 360*x^5 + 2040*x^6 + ...
%e A002898 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
%p A002898 a:= proc(n) option remember; `if`(n<3, [1, 0, 6][n+1], ((n-1)*
%p A002898       n*a(n-1) +24*(n-1)^2*a(n-2) +36*(n-1)*(n-2)*a(n-3))/n^2)
%p A002898     end:
%p A002898 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 08 2020
%t A002898 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 *)
%o A002898 (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 */
%Y A002898 Cf. A000172, A006480, A337905-A337907, A094060, A002894 (returns square lattice), A002893 (honeycomb net).
%K A002898 nonn,walk,nice
%O A002898 0,3
%A A002898 _N. J. A. Sloane_
%E A002898 More terms from David Bloom, Mar 1997
%E A002898 Formula and further terms from _Cyril Banderier_, Oct 12 2000