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A342511 Irregular triangle read by rows: T(n,k) is the number of substrings of n read in binary that are instances of the Zimin word Z_k. 1 <= n <= A342510(n). 4
1, 1, 3, 3, 6, 6, 1, 6, 6, 1, 10, 1, 10, 1, 10, 2, 10, 2, 10, 10, 2, 10, 1, 10, 3, 15, 3, 15, 2, 15, 4, 15, 2, 15, 3, 15, 4, 15, 4, 15, 4, 15, 1, 15, 2, 15, 3, 15, 4, 15, 1, 15, 4, 15, 3, 15, 6, 21, 6, 21, 4, 21, 6, 21, 3, 21, 6, 21, 6, 21, 5, 21, 4, 21, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Peter Kagey, Table of n, a(n) for n = 0..10170 (first 2^12 rows)

Peter Kagey, Matching ABACABA-type patterns, Code Golf Stack Exchange.

Danny Rorabaugh, Toward the Combinatorial Limit Theory of Free Words, arXiv preprint arXiv:1509.04372 [math.CO], 2015.

Wikipedia, Sesquipower.

FORMULA

T(n,1) = A000217(A070939(n)).

EXAMPLE

   n | binary  |k=1,  2, 3

-----+---------+----------

   0 |       0 |  1

   1 |       1 |  1

   2 |      10 |  3

   3 |      11 |  3

   4 |     100 |  6

   5 |     101 |  6,  1

   6 |     110 |  6

   7 |     111 |  6,  1

   8 |    1000 | 10,  1

   9 |    1001 | 10,  1

  10 |    1010 | 10,  2

  11 |    1011 | 10,  2

  12 |    1100 | 10

  13 |    1101 | 10,  2

  14 |    1110 | 10,  1

  15 |    1111 | 10,  3

  16 |   10000 | 15,  3

           ...

  85 | 1010101 | 28, 11, 1

For n = 121, the binary expansion is "1111001", which has 28 nonempty substrings.

For k = 1, there are T(121,1) = 28 substrings that are instances of Z_1 = A.

For k = 2, there are T(121,2) = 7 substrings that are instances of Z_2 = ABA are:

(111)1001 with A = 1 and B = 1,

1(111)001 with A = 1 and B = 1,

(1111)001 with A = 1 and B = 11,

111(1001) with A = 1 and B = 00,

11(11001) with A = 1 and B = 100,

1(111001) with A = 1 and B = 1100, and

(1111001) with A = 1 and B = 11100.

CROSSREFS

Cf. A000217, A070939, A342510, A342512.

Sequence in context: A038167 A307967 A131077 * A175520 A271668 A072464

Adjacent sequences:  A342508 A342509 A342510 * A342512 A342513 A342514

KEYWORD

nonn,base,tabf,look

AUTHOR

Peter Kagey, Mar 14 2021

STATUS

approved

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Last modified October 27 21:44 EDT 2021. Contains 348290 sequences. (Running on oeis4.)