A342511 - OEIS

%I #21 Mar 20 2021 14:41:03

%S 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,

%T 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,

%U 6,21,6,21,4,21,6,21,3,21,6,21,6,21,5,21,4,21,5

%N 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).

%H Peter Kagey, <a href="/A342511/b342511.txt">Table of n, a(n) for n = 0..10170</a> (first 2^12 rows)

%H Peter Kagey, <a href="https://codegolf.stackexchange.com/q/220679/53884">Matching ABACABA-type patterns</a>, Code Golf Stack Exchange.

%H Danny Rorabaugh, <a href="http://arxiv.org/abs/1509.04372">Toward the Combinatorial Limit Theory of Free Words</a>, arXiv preprint arXiv:1509.04372 [math.CO], 2015.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sesquipower">Sesquipower</a>.

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

%e    n | binary  |k=1,  2, 3

%e -----+---------+----------

%e    0 |       0 |  1

%e    1 |       1 |  1

%e    2 |      10 |  3

%e    3 |      11 |  3

%e    4 |     100 |  6

%e    5 |     101 |  6,  1

%e    6 |     110 |  6

%e    7 |     111 |  6,  1

%e    8 |    1000 | 10,  1

%e    9 |    1001 | 10,  1

%e   10 |    1010 | 10,  2

%e   11 |    1011 | 10,  2

%e   12 |    1100 | 10

%e   13 |    1101 | 10,  2

%e   14 |    1110 | 10,  1

%e   15 |    1111 | 10,  3

%e   16 |   10000 | 15,  3

%e            ...

%e   85 | 1010101 | 28, 11, 1

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

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

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

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

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

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

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

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

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

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

%Y Cf. A000217, A070939, A342510, A342512.

%K nonn,base,tabf,look

%O 0,3

%A _Peter Kagey_, Mar 14 2021