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 A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
 1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017 LINKS Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc. Square array A(n,k) begins:   1,  1,  1,  1,  1,   1,   1,   1, ...   0,  1,  2,  3,  4,   5,   6,   7, ...   0,  0,  2,  4,  6,   8,  10,  12, ...   0,  0,  2,  6, 10,  14,  18,  22, ...   0,  0,  2,  8, 16,  24,  32,  40, ...   0,  0,  2, 12, 26,  42,  58,  74, ...   0,  0,  2, 16, 42,  72, 104, 136, ...   0,  0,  2, 24, 68, 126, 188, 252, ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1,       `if`(i=0, add(b(n-1, j, k), j=1..k),       `if`(i>1, b(n-1, i-1, k), 0)+       `if`(i b(n, 0, k): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[i1, (i, j)->abs(i-j)==1, j->1, z); a(n, k)=Vec(ColGf(k, x) + O(x^(n+1)))[n+1]; for(n=0, 7, for(k=0, 7, print1( a(n, k), ", ") ); print(); ); \\ Andrew Howroyd, Apr 17 2017 CROSSREFS Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360. Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4. Main diagonal gives: A102699. Cf. A198632, A188866, A276562, A208727, A208671. Sequence in context: A292377 A216238 A157608 * A216054 A217257 A217315 Adjacent sequences:  A220059 A220060 A220061 * A220063 A220064 A220065 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 03 2012 STATUS approved

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Last modified May 30 09:21 EDT 2020. Contains 334717 sequences. (Running on oeis4.)