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A337905 The number of walks of n steps on the hexagonal lattice that start at the origin and end at the adjacent vertex (1,0). 4
1, 2, 15, 60, 340, 1680, 9135, 48440, 264726, 1446060, 7996296, 44396352, 248133600, 1392623232, 7850732175, 44413669872, 252098234674, 1435074678180, 8190821465970, 46860693370920, 268676908816680, 1543504863288960, 8883248453674920, 51210412534906560 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017, see W_{xyzn}^{hon}.
FORMULA
D-finite with recurrence (n+1)^2*a(n) -n*(n+1)*a(n-1) -24n^2*a(n-2) -36*n*(n-1)*a(n-3)=0.
a(n+1) = A002898(n)+2*a(n)+2*A337907(n)+A337906(n).
EXAMPLE
There are a(2)=2 paths with 2 steps: UD or DU, where R=(1,0), L=(-1,0), U=(1/2,sqrt(3)/2), u=(-1/2,sqrt(3)/2), D=(1/2,-sqrt(3)/2), d=(-1/2,-sqrt(3)/2).
There are a(3)=15 paths with 3 steps: 6 paths permutations of RuD, 6 permutations of RUd, and 3 permutations of RRL.
MAPLE
HexLat := proc(n, finx, finy)
local a, L, R;
a := 0 ;
for L from 0 to n do
for R from modp(n+finy-L, 2) to n-L by 2 do
a := a+ binomial(n, L) *binomial(n-L, R) *binomial(n-L-R, n/2+L/2-3*R/2+finx) *binomial(n-L-R, (n-L-R-finy)/2) ;
end do:
end do:
a ;
end proc:
seq(HexLat(n, 0, 0), n=0..15) ; # A002898
seq(HexLat(n, 1, 0), n=0..15) ; # A337905
seq(HexLat(n, 2, 0), n=0..15) ; # A337906
seq(HexLat(n, 1/2, 1), n=0..15) ; # A337905
MATHEMATICA
HexLat[n_, finx_, finy_] := Module[{a = 0, L, R}, For[L = 0, L <= n, L++, For[R = Mod[n + finy - L, 2], R <= n - L, R += 2, a = a + Binomial[n, L]*Binomial[n - L, R]*Binomial[n - L - R, n/2 + L/2 - 3*R/2 + finx]*Binomial[n - L - R, (n - L - R - finy)/2]]]; a];
Table[HexLat[n, 1, 0], {n, 1, 24}] (* Jean-François Alcover, Jun 25 2023, after R. J. Mathar *)
CROSSREFS
Cf. A002898 (returns to origin), A337906, A337907.
Sequence in context: A071237 A006470 A084169 * A296661 A000181 A047146
KEYWORD
nonn,walk
AUTHOR
R. J. Mathar, Sep 29 2020
STATUS
approved

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Last modified March 29 05:28 EDT 2024. Contains 371264 sequences. (Running on oeis4.)