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A285281
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Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 3 or less.
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4
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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
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OFFSET
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4,2
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COMMENTS
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All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.
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LINKS
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EXAMPLE
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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 ...
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MATHEMATICA
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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 *)
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PROG
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(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(); );
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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