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.
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
Offset: 0
Examples
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, ...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
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.
Programs
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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 -
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[i -
PARI
TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i))); ColGf(m,z)=1+z*TransferGf(m, i->1, (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
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