A338672 Number of n-step closed walks on the kagomé lattice.

1, 0, 4, 4, 28, 60, 264, 784, 3004, 10204, 37824, 135784, 502784, 1851200, 6901696, 25766144, 96797244, 364655100, 1379120400, 5230011896, 19890313128, 75823622984, 289698620336, 1109059301536, 4253731156128, 16342545417760, 62885474132992, 242331022479040, 935085717105792, 3612737418620032, 13974224404904704

Offset

0,3

Links

Table of n, a(n) for n=0..30.

Li Gan, Algebraic Area of Lattice Random Walks and Exclusion Statistics, PhD thesis, Université Paris-Saclay, 2023. (see Appendix D.4)

B. Helffer, P. Kerdelhué, and J. Royo-Letelier, Chambers's formula for the graphene and the Hou model with kagome periodicity and applications, arXiv:1408.2814 [math.AP], 2014; Ann. Henri Poincaré, 17 (2016), 795-818.

P. Kerdelhué and J. Royo-Letelier, On the low lying spectrum of the magnetic Schrödinger operator with kagome periodicity, arXiv:1404.0642 [math.AP], 2014; Rev. Math. Phys., 26 (2014), 1450020.

Wikipedia, Trihexagonal tiling.

Formula

a(n) is the constant coefficient in the expansion of 1/3 * trace(A^n), where A is the matrix {{0, y+x*y, y+1/x}, {1/y+1/(x*y), 0, 1/x+1/(x*y)}, {x+1/y, x+x*y, 0}}.

a(n) = (1/3) * ((-2)^n + 2 * Sum_{k=0..floor(n/2)} Sum_{j=0..k} binomial(n, 2*k) * binomial(k, j)^2 * binomial(2*j, j)).

a(n) ~ 2^(2*n+1) / (Pi*sqrt(3)*n). - Vaclav Kotesovec, Oct 10 2024

Mathematica

a[n_] := ((-2)^n + 2 Sum[Binomial[n, 2 k] Binomial[k, j]^2 Binomial[2 j, j], {k, 0, Floor[n/2]}, {j, 0, k}])/3; Table[a[n], {n, 0, 30}]

Prog

(PARI) a(n)={((-2)^n + 2 * sum(k=0, n\2, sum(j=0, k, binomial(n, 2*k) * binomial(k, j)^2 * binomial(2*j, j))))/3} \\ Andrew Howroyd, Apr 24 2021

Crossrefs

Cf. A005397, A001665.

Sequence in context: A227715 A173049 A272040 * A065237 A264586 A348635

Adjacent sequences: A338669 A338670 A338671 * A338673 A338674 A338675

Keyword

nonn,walk

Author

Li GAN and Stéphane Ouvry, Apr 23 2021

Status

approved