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%I #8 Jun 16 2017 22:33:29
%S 1,4,1,16,5,1,64,23,6,1,256,101,30,7,1,1024,467,138,37,8,1,4096,2165,
%T 694,175,44,9,1,16384,10055,3526,925,212,51,10,1,65536,46709,18012,
%U 4977,1156,249,58,11,1,262144,216995,92140,27067,6428,1387,286,65,12,1
%N Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 3 or less.
%C All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.
%H Andrew Howroyd, <a href="/A285281/b285281.txt">Table of n, a(n) for n = 4..1278</a>
%e Table starts (m>=4, n>=0):
%e 1 4 16 64 256 1024 4096 16384 65536 ...
%e 1 5 23 101 467 2165 10055 46709 216995 ...
%e 1 6 30 138 694 3526 18012 92140 471566 ...
%e 1 7 37 175 925 4977 27067 147777 808165 ...
%e 1 8 44 212 1156 6428 36338 206942 1183164 ...
%e 1 9 51 249 1387 7879 45663 267367 1575395 ...
%e 1 10 58 286 1618 9330 54994 328058 1973026 ...
%e 1 11 65 323 1849 10781 64325 388749 2371457 ...
%e 1 12 72 360 2080 12232 73656 449440 2770016 ...
%t diff = 3; m0 = diff + 1; mmax = 13;
%t TransferGf[m_, u_, t_, v_, z_] := Array[u, m].LinearSolve[IdentityMatrix[m] - z*Array[t, {m, m}], Array[v, m]]
%t RowGf[d_, m_, z_] := 1 + z*Sum[TransferGf[m, Boole[# == k] &, Boole[Abs[#1 - #2] <= d] &, Boole[Abs[# - k] <= d] &, z], {k, 1, m}];
%t row[m_] := row[m] = CoefficientList[RowGf[diff, m, x] + O[x]^mmax, x];
%t T[m_ /; m >= m0, n_ /; n >= 0] := row[m][[n + 1]];
%t Table[T[m - n , n], {m, m0, mmax}, {n, m - m0, 0, -1}] // Flatten (* _Jean-François Alcover_, Jun 16 2017, adapted from PARI *)
%o (PARI)
%o 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)));
%o 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));
%o for(m=4, 12, print(RowGf(3,m,x)));
%o for(m=4, 12, v=Vec(RowGf(3,m,x) + O(x^9)); for(n=1, length(v), print1( v[n], ", ") ); print(); );
%Y Rows 5-32 are A124999, A125316-A125342.
%Y Cf. A285267, A285280, A276562.
%K nonn,tabl
%O 4,2
%A _Andrew Howroyd_, Apr 15 2017
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