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A002898
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Number of n-step closed paths on hexagonal lattice.
(Formerly M4101 N1701)
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3
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1, 0, 6, 12, 90, 360, 2040, 10080, 54810, 290640, 1588356, 8676360, 47977776, 266378112, 1488801600, 8355739392, 47104393050, 266482019232, 1512589408044, 8610448069080, 49144928795820, 281164160225520
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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.
Contribution from David Callan (callan(AT)stat.wisc.edu), Aug 25 2009: (Start)
a(n) = number of 2-by-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
12..13..21..23..31..32
21..31..12..32..13..23. (End)
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REFERENCES
| C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
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.
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).
Cf. solution to 1995 Putnam problem A-6, Am. Math. Monthly, 1996, p. 674.
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LINKS
| C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
| a(0) = 1, a(1) = 0, a(2)=6, (108*n+72+36*n^2)*a(n)+(24*n^2+96*n+96)*a(n+1)+(n^2+5*n+6)*a(n+2)+(-6*n-9-n^2)*a(n+3)=0.
E.g.f.: (BesselI(0,2*x))^3+2*sum((BesselI(k,2*x))^3,k=1..infinity), from Karol A. Penson (penson(AT)lptl.jussieu.fr) Aug 18 2006.
a(n)=\sum_{i=0}^{n} (-2)^{n-i}\binom{n}{i}(\sum_{j=0}^{i} \binom{i}{j}^{3}) [From 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
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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}] (* From Jean-François Alcover, Dec 21 2011, after Vasu Tewari *)
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CROSSREFS
| Sequence in context: A005402 A128953 A181597 * A003613 A099767 A191462
Adjacent sequences: A002895 A002896 A002897 * A002899 A002900 A002901
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KEYWORD
| nonn,walk,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from David Bloom 3/97.
Formula and further terms from Cyril Banderier (Cyril.Banderier(AT)inria.fr), Oct 12 2000
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