A319411 Triangle read by rows: T(n,k) = number of binary vectors of length n with runs-resistance k (1 <= k <= n).

2, 2, 2, 2, 2, 4, 2, 4, 6, 4, 2, 2, 12, 12, 4, 2, 6, 30, 18, 8, 0, 2, 2, 44, 44, 32, 4, 0, 2, 6, 82, 76, 74, 16, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0, 2, 10, 790, 752, 1500, 938, 104, 0, 0, 0, 0, 0

Offset

1,1

Comments

"Runs-resistance" is defined in A318928.

Row sums are 2,4,8,16,... (the binary vectors may begin with 0 or 1).

This is similar to A329767 but without the k = 0 column and with a different row n = 1. - Gus Wiseman, Nov 25 2019

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..1830

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 Apparently unpublished.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.

Example

Triangle begins:
2,
2, 2,
2, 2, 4,
2, 4, 6, 4,
2, 2, 12, 12, 4,
2, 6, 30, 18, 8, 0,
2, 2, 44, 44, 32, 4, 0,
2, 6, 82, 76, 74, 16, 0, 0,
2, 4, 144, 138, 172, 52, 0, 0, 0,
2, 6, 258, 248, 350, 156, 4, 0, 0, 0,
2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0,
2, 10, 790, 752, 1500, 938, 104, 0, 0, 0, 0, 0,
...
Lenormand gives the first 20 rows.
The calculation of row 4 is as follows.
We may assume the first bit is a 0, and then double the answers.
vector / runs / steps to reach a single number:
0000 / 4 / 1
0001 / 31 -> 11 -> 2 / 3
0010 / 211 -> 12 -> 11 -> 2 / 4
0011 / 22 -> 2 / 2
0100 / 112 -> 21 -> 11 -> 2 / 4
0101 / 1111 -> 4 / 2
0110 / 121 -> 111 -> 3 / 3
0111 / 13 -> 11 -> 2 / 3
and we get 1 (once), 2 (twice), 3 (three times) and 4 (twice).
So row 4 is: 2,4,6,4.

Mathematica

runsresist[q_]:=If[Length[q]==1, 1, Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1];
Table[Length[Select[Tuples[{0, 1}, n], runsresist[#]==k&]], {n, 10}, {k, n}] (* Gus Wiseman, Nov 25 2019 *)

Crossrefs

Row sums are A000079.

Column k = 2 is 2 * A032741 = A319410.

Column k = 3 is 2 * A329745 (because runs-resistance 2 for compositions corresponds to runs-resistance 3 for binary words).

The version for compositions is A329744.

The version for partitions is A329746.

The number of nonzero entries in row n > 0 is A319412(n).

The runs-resistance of the binary expansion of n is A318928.

Cf. A096365, A225485, A245563, A319413, A319414, A319415, A325280, A329747, A329750, A329767, A329870.

Sequence in context: A086876 A066691 A260123 * A164296 A319817 A233566

Adjacent sequences: A319408 A319409 A319410 * A319412 A319413 A319414

Keyword

nonn,tabl,base

Author

N. J. A. Sloane, Sep 20 2018

Status

approved