A282715 -id:A282715 - cp's OEIS frontend

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

Manon Stipulanti, Mar 27 2017

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.

			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.

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.

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

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

A284442 Number of nonzero terms in first n rows of the base-3 generalized Pascal triangle P_3 (A284441).

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 18, 22, 25, 29, 34, 40, 45, 49, 55, 62, 69, 75, 79, 85, 90, 97, 103, 110, 115, 121, 125, 130, 137, 145, 153, 160, 170, 180, 191, 200, 207, 215, 225, 232, 237, 245, 256, 266, 275, 285, 298, 310, 323

Offset: 0

Manon Stipulanti, Mar 27 2017

nonn

Manon Stipulanti, Table of n, a(n) for n = 0..6560
Julien Leroy, Michel Rigo, and Manon Stipulanti, Behavior of Digital Sequences Through Exotic Numeration Systems, Electronic Journal of Combinatorics 24(1) (2017), #P1.44. See also.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.
Julien Leroy, Michel Rigo, and 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.

Partial sums of A282715.

Cf. A284441.

cp's OEIS Frontend

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

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