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A285280 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 2 or less. 4
1, 3, 1, 9, 4, 1, 27, 14, 5, 1, 81, 46, 19, 6, 1, 243, 162, 65, 24, 7, 1, 729, 574, 247, 84, 29, 8, 1, 2187, 2042, 955, 332, 103, 34, 9, 1, 6561, 7270, 3733, 1336, 417, 122, 39, 10, 1, 19683, 25890, 14649, 5478, 1717, 502, 141, 44, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

3,2

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 = 3..1277

EXAMPLE

Table starts (m>=3, n>=0):

1  3  9  27  81  243   729  2187 ...

1  4 14  46 162  574  2042  7270 ...

1  5 19  65 247  955  3733 14649 ...

1  6 24  84 332 1336  5478 22658 ...

1  7 29 103 417 1717  7229 30793 ...

1  8 34 122 502 2098  8980 38928 ...

1  9 39 141 587 2479 10731 47063 ...

1 10 44 160 672 2860 12482 55198 ...

MATHEMATICA

diff = 2; m0 = diff + 1; mmax = 12;

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=3, 10, print(RowGf(2, m, x)));

for(m=3, 10, v=Vec(RowGf(2, m, x) + O(x^8)); for(n=1, length(v), print1( v[n], ", ") ); print(); );

CROSSREFS

Rows 4-32 are A124805, A124806, A124807, A124828, A124843, A124851, A124852, A124857, A124858, A124864, A124892-A124894, A124898, A124935, A124947, A124948-A124958, A124994, A124998.

Cf. A285266, A276562, A285281.

Sequence in context: A054448 A106516 A140071 * A285266 A067417 A187887

Adjacent sequences:  A285277 A285278 A285279 * A285281 A285282 A285283

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 15 2017

STATUS

approved

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Last modified December 6 13:45 EST 2021. Contains 349563 sequences. (Running on oeis4.)