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A212334 Number of words, either empty or beginning with the first letter of the 4-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet. 2
1, 1, 9, 163, 3593, 87501, 2266155, 61211095, 1704838665, 48605519665, 1411522695509, 41606511550803, 1241591466423467, 37435593955828069, 1138713916992923679, 34901292375152457663, 1076813644170756916745, 33416749492077957930105, 1042376218505671236116985 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also the number of (4*n-1)-step walks on 4-dimensional cubic lattice from (1,0,0,0) to (n,n,n,n) with positive unit steps in all dimensions such that the absolute difference of the dimension indices used in consecutive steps is <= 1.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

FORMULA

a(n) ~ 2*(2^(3/4)-2^(1/4)) * (17+12*sqrt(2))^n/(4*Pi*n)^(3/2). - Vaclav Kotesovec, Aug 13 2013

MAPLE

a:= proc(n) option remember; `if`(n<3, [1, 1, 9][n+1],

      ((26682*n^4 -102687*n^3 +149385*n^2 -109413*n +31101) *a(n-1)

      +(-161058*n^4 +1392915*n^3 -4418826*n^2 +6030348*n -2931516) *a(n-2)

      +(4718*n^4 -47957*n^3 +176841*n^2 -275751*n +148365) *a(n-3)) /

      (n^3 *(646*n -1057)))

    end:

seq(a(n), n=0..30);

MATHEMATICA

a[n_] := a[n] = If[n < 3, {1, 1, 9}[[n + 1]], ((26682 n^4 - 102687 n^3 + 149385 n^2 - 109413 n + 31101) a[n-1] + (-161058 n^4 + 1392915 n^3 - 4418826 n^2 + 6030348 n - 2931516)a[n-2] + (4718 n^4 - 47957 n^3 + 176841 n^2 - 275751 n + 148365)a[n-3])/(n^3 (646 n - 1057))];

a /@ Range[0, 30] (* Jean-Fran├žois Alcover, May 14 2020, after Maple *)

CROSSREFS

Column k=4 of A208673.

Sequence in context: A297436 A041147 A041144 * A086759 A053130 A219074

Adjacent sequences:  A212331 A212332 A212333 * A212335 A212336 A212337

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 07 2012

STATUS

approved

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Last modified May 9 19:28 EDT 2021. Contains 343746 sequences. (Running on oeis4.)