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A276562 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 1 or less. 5
1, 1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 10, 5, 1, 32, 35, 22, 13, 6, 1, 64, 83, 54, 29, 16, 7, 1, 128, 199, 134, 73, 36, 19, 8, 1, 256, 479, 340, 185, 92, 43, 22, 9, 1, 512, 1155, 872, 481, 236, 111, 50, 25, 10, 1, 1024, 2787, 2254, 1265, 622, 287, 130, 57, 28, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 = 1..1275

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

FORMULA

T(m, n) = Sum_{j=1..m} (1 + 2*cos(j*pi/(m+1)))^n. - Andrew Howroyd, Apr 15 2017

EXAMPLE

Array starts:

   1  1  1   1   1    1    1    1     1     1 ...

   2  4  8  16  32   64  128  256   512  1024 ...

   3  7 15  35  83  199  479 1155  2787  6727 ...

   4 10 22  54 134  340  872 2254  5854 15250 ...

   5 13 29  73 185  481 1265 3361  8993 24193 ...

   6 16 36  92 236  622 1658 4468 12132 33146 ...

   7 19 43 111 287  763 2051 5575 15271 42099 ...

   8 22 50 130 338  904 2444 6682 18410 51052 ...

   9 25 57 149 389 1045 2837 7789 21549 60005 ...

  10 28 64 168 440 1186 3230 8896 24688 68958 ...

MATHEMATICA

T[m_, n_] := Sum[(1 + 2*Cos[j*Pi/(m+1)])^n, {j, 1, m}] // FullSimplify;

Table[T[m-n+1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* Jean-Fran├žois Alcover, Jun 06 2017 *)

PROG

(PARI) \\ from Knopfmacher et al.

ChebyshevU(n, x) = sum(i=0, n/2, 2*poltchebi(n-2*i, x)) + (n%2-1);

RowGf(k, x) = 1 + (k*x*(1+3*x) - 2*(k+1)*x*subst(ChebyshevU(k-1, z)/ChebyshevU(k, z), z, (1-x)/(2*x)))/((1+x)*(1-3*x));

a(m, n)=Vec(RowGf(m, x)+O(x^(n+1)))[n+1];

for(m=1, 10, print(RowGf(m, x)));

for(m=1, 10, for(n=1, 9, print1( a(m, n), ", ") ); print(); );

CROSSREFS

Rows 3-32 are A124696-A124719, A124726, A124783, A124784, A124799, A124803, A124804.

Cf. A188866, A220062, A285280, A285281, A208777, A208721.

Sequence in context: A109435 A134392 A048483 * A055248 A103316 A140069

Adjacent sequences:  A276559 A276560 A276561 * A276563 A276564 A276565

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 15 2017

STATUS

approved

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)