A284441 Base-3 generalized Pascal triangle P_3 read by rows (see Comments for precise definition).

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, 1, 2, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Author

Manon Stipulanti, Mar 27 2017

Keywords

• nonn
• tabl

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.

Examples

			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.
		

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.

Crossrefs

Cf. A282715 (gives (essentially) the number of nonzero entries in the rows), A284442 (their partial sums).

Programs

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