A319421 Triangle read by rows: T(n,k) (1 <= k <= n) = one-half of the number of binary vectors of length n and cuts-resistance k.

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 6, 2, 1, 1, 10, 11, 7, 2, 1, 1, 14, 24, 14, 8, 2, 1, 1, 22, 42, 35, 16, 9, 2, 1, 1, 30, 81, 68, 45, 18, 10, 2, 1, 1, 46, 138, 149, 89, 55, 20, 11, 2, 1, 1, 62, 250, 282, 216, 110, 66, 22, 12, 2, 1, 1, 94, 419, 577, 422, 285, 132, 78, 24, 13, 2, 1

Offset

1,5

Comments

Cuts-resistance is defined in A319416.

This triangle summarizes the data shown in A319420.

Conjecture (Sloane): Sum_{i = 1..n} i * T(n,i) = A189391(n).

Links

Table of n, a(n) for n=1..78.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. See page 4.

N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Formula

T(n,k) = A329860(n,k)/2. - Gus Wiseman, Nov 25 2019

Example

Triangle begins:
   1
   1   1
   1   2   1
   1   4   2   1
   1   6   6   2   1
   1  10  11   7   2   1
   1  14  24  14   8   2   1
   1  22  42  35  16   9   2   1
   1  30  81  68  45  18  10   2   1
   1  46 138 149  89  55  20  11   2   1
   1  62 250 282 216 110  66  22  12   2   1
   1  94 419 577 422 285 132  78  24  13   2   1
Lenormand gives first 15 rows.
For example, the "1,2,1" row here refers to the 8 vectors of length 3. There are 2 vectors of cuts-resistance 1, namely 010 and 101 (see A319416), 4 vectors of cuts-resistance 2 (100,011,001,110), and 2 of cuts-resistance 3 (000 and 111). Halving these counts we get 1,2,1

Mathematica

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[Length[Select[Tuples[{0, 1}, n], First[#]==1&&degdep[#]==k&]], {n, 8}, {k, n}] (* Gus Wiseman, Nov 25 2019 *)

Crossrefs

Row sums are A000079.

Column k = 2 appears to be A027383.

The version for runs-resistance is A319411 or A329767.

The version for compositions is A329861.

The cuts-resistance of the binary expansion of n is A319416(n).

Cf. A000975, A164707, A189391, A261983, A318921, A318928, A319420, A329738, A329860, A329865.

Sequence in context: A211970 A089688 A229706 * A092479 A124022 A098063

Adjacent sequences: A319418 A319419 A319420 * A319422 A319423 A319424

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 23 2018

Status

approved