OFFSET
0,12
COMMENTS
List the base-3 numbers in their natural order as base-3 strings, beginning with the empty string epsilon, which represents 0. Row n of the triangle gives the number of times the k-th string occurs as a (scattered) substring of the n-th string.
LINKS
Manon Stipulanti, Table of n, a(n) for n = 0..29645
J. Leroy, M. Rigo, and M. Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Mathematics 340 (2017), 862-881.
Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.
Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.
EXAMPLE
Triangle begins:
1,
1, 1,
1, 0, 1,
1, 1, 0, 1,
1, 2, 0, 0, 1,
1, 1, 1, 0, 0, 1,
1, 0, 1, 0, 0, 0, 1
1, 1, 1, 0, 0, 0, 0, 1,
1, 0, 2, 0, 0, 0, 0, 0, 1,
1, 1, 0, 2, 0, 0, 0, 0, 0, 1,
1, 2, 0, 1, 1, 0, 0, 0, 0, 0, 1,
1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1,
1, 2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1,
1, 3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1
...
The base-3 numbers are epsilon, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 220, 221, 222, ... The tenth number 101 contains
eps 1 2 10 11 12 20 21 22 100 101 respectively
.1..2.0..1..1..0..0..0..0..0...1 times, which is row 10 of the triangle.
MATHEMATICA
coeff[u_, v_] := coeff[u, v] = If[Length[v] == 0, 1, If[Length[u] < Length[v], 0, coeff[Drop[u, -1], v] + ((Last[u] == Last[v]) /. {True -> 1, False -> 0}) coeff[Drop[u, -1], Drop[v, -1]]]]
P3 = Table[coeff[IntegerDigits[i, 3] /. {0} -> {}, IntegerDigits[j, 3] /. {0} -> {}], {i, 0, 3^5 - 1}, {j, 0, i}] //Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Manon Stipulanti, Mar 27 2017
STATUS
approved