A285281 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 3 or less.

1, 4, 1, 16, 5, 1, 64, 23, 6, 1, 256, 101, 30, 7, 1, 1024, 467, 138, 37, 8, 1, 4096, 2165, 694, 175, 44, 9, 1, 16384, 10055, 3526, 925, 212, 51, 10, 1, 65536, 46709, 18012, 4977, 1156, 249, 58, 11, 1, 262144, 216995, 92140, 27067, 6428, 1387, 286, 65, 12, 1

Offset

4,2

Comments

All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.

Links

Andrew Howroyd, Table of n, a(n) for n = 4..1278

Example

Table starts (m>=4, n>=0):
1  4 16  64  256  1024  4096  16384   65536 ...
1  5 23 101  467  2165 10055  46709  216995 ...
1  6 30 138  694  3526 18012  92140  471566 ...
1  7 37 175  925  4977 27067 147777  808165 ...
1  8 44 212 1156  6428 36338 206942 1183164 ...
1  9 51 249 1387  7879 45663 267367 1575395 ...
1 10 58 286 1618  9330 54994 328058 1973026 ...
1 11 65 323 1849 10781 64325 388749 2371457 ...
1 12 72 360 2080 12232 73656 449440 2770016 ...

Mathematica

diff = 3; m0 = diff + 1; mmax = 13;
TransferGf[m_, u_, t_, v_, z_] := Array[u, m].LinearSolve[IdentityMatrix[m] - z*Array[t, {m, m}], Array[v, m]]
RowGf[d_, m_, z_] := 1 + z*Sum[TransferGf[m, Boole[# == k] &, Boole[Abs[#1 - #2] <= d] &, Boole[Abs[# - k] <= d] &, z], {k, 1, m}];
row[m_] := row[m] = CoefficientList[RowGf[diff, m, x] + O[x]^mmax, x];
T[m_ /; m >= m0, n_ /; n >= 0] := row[m][[n + 1]];
Table[T[m - n , n], {m, m0, mmax}, {n, m - m0, 0, -1}] // Flatten (* Jean-François Alcover, Jun 16 2017, adapted from PARI *)

Prog

(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)));
RowGf(d, m, z)=1+z*sum(k=1, m, TransferGf(m, i->if(i==k, 1, 0), (i, j)->abs(i-j)<=d, j->if(abs(j-k)<=d, 1, 0), z));
for(m=4, 12, print(RowGf(3, m, x)));
for(m=4, 12, v=Vec(RowGf(3, m, x) + O(x^9)); for(n=1, length(v), print1( v[n], ", ") ); print(); );

Crossrefs

Rows 5-32 are A124999, A125316-A125342.

Cf. A285267, A285280, A276562.

Sequence in context: A167343 A094361 A187926 * A285267 A067425 A188481

Adjacent sequences: A285278 A285279 A285280 * A285282 A285283 A285284

Keyword

nonn,tabl

Author

Andrew Howroyd, Apr 15 2017

Status

approved