|
|
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.
|
|
6
|
|
|
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
|
|
|
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;
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|