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A285267 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with adjacent elements differing by 3 or less.

%I A285267 #10 Jun 17 2017 02:29:00
%S A285267 1,4,1,16,5,1,64,23,6,1,256,107,30,7,1,1024,497,154,37,8,1,4096,2309,
%T A285267 788,203,44,9,1,16384,10727,4034,1111,252,51,10,1,65536,49835,20650,
%U A285267 6083,1446,301,58,11,1,262144,231521,105708,33305,8300,1787,350,65,12,1
%N A285267 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with adjacent elements differing by 3 or less.
%C A285267 All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.
%H A285267 Andrew Howroyd, <a href="/A285267/b285267.txt">Table of n, a(n) for n = 4..1278</a>
%e A285267 Array starts (m>=4, n>=0):
%e A285267 1  4 16  64  256  1024  4096  16384 ...
%e A285267 1  5 23 107  497  2309 10727  49835 ...
%e A285267 1  6 30 154  788  4034 20650 105708 ...
%e A285267 1  7 37 203 1111  6083 33305 182349 ...
%e A285267 1  8 44 252 1446  8300 47642 273466 ...
%e A285267 1  9 51 301 1787 10619 63111 375091 ...
%e A285267 1 10 58 350 2130 12990 79258 483646 ...
%e A285267 1 11 65 399 2473 15381 95757 596341 ...
%t A285267 diff = 3; m0 = 4; mmax = 13;
%t A285267 TransferGf[m_, u_, t_, v_, z_] := Array[u, m].LinearSolve[IdentityMatrix[m] - z*Array[t, {m, m}], Array[v, m]]
%t A285267 RowGf[d_, m_, z_] := 1+z*TransferGf[m, 1&, Boole[Abs[#1-#2] <= d]&, 1&, z];
%t A285267 row[m_] := row[m] = CoefficientList[RowGf[diff, m, x] + O[x]^mmax, x];
%t A285267 T[m_ /; m >= m0, n_ /; n >= 0] := row[m][[n + 1]];
%t A285267 Table[T[m - n, n], {m, m0, mmax}, {n, m - m0, 0, -1}] // Flatten (* _Jean-François Alcover_, Jun 17 2017, adapted from PARI *)
%o A285267 (PARI)
%o A285267 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 A285267 RowGf(d,m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)<=d, j->1, z);
%o A285267 for(m=4, 12, print(RowGf(3,m,x)));
%o A285267 for(m=4, 12, v=Vec(RowGf(3,m,x) + O(x^9)); for(n=1, length(v), print1( v[n], ", ") ); print(); );
%Y A285267 Rows 5-32 are A126473-A126500.
%Y A285267 Cf. A285281, A188866, A285266.
%K A285267 nonn,tabl
%O A285267 4,2
%A A285267 _Andrew Howroyd_, Apr 15 2017