OFFSET
0,3
COMMENTS
T(n,k) is the number of occurrences of the periodic substring (01)^k in the periodic string (0011)^n (see Proposition 4.3 at page 6 in Fang).
The word (w_1, w_2, ..., w_r)^m is defined as the word obtained by concatenating (w_1, w_2, ..., w_r) m times.
A word w' = (w'_1, w'_2, ..., w'_s) is said be a subword of a given word w = (w_1, w_2, ..., w_r), if there is some set P = {p_1 < ... < p_s} of integers from 1 to r satisfying w_{p_j} = w'_j for all 1 <= j <= s, and we call the set P an occurrence of w' in w (see Preliminaries section at pp. 2-3 in Fang).
LINKS
Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024.
FORMULA
EXAMPLE
The triangle begins as:
1;
1, 4;
1, 12, 16;
1, 24, 80, 64;
1, 40, 240, 448, 256;
1, 60, 560, 1792, 2304, 1024;
1, 84, 1120, 5376, 11520, 11264, 4096;
...
T(2,1) = 12 since there are 12 occurrences of (01)^1 = 01 in (0011)^2 = 00110011: {1, 3}, {1, 4}, {1, 7}, {1, 8}, {2, 3}, {2, 4}, {2, 7}, {2, 8}, {5, 7}, {5, 8}, {6, 7}, {6, 8}.
MATHEMATICA
T[n_, k_]:=4^k Binomial[n+k, n-k]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* or *)
T[n_, k_]:=SeriesCoefficient[(1-x)/((1-x)^2-4x y), {x, 0, n}, {y, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, Jun 09 2024
STATUS
approved