A279026 Size of blocks of 1's in the second column of Catalan numbers written in binary and left-aligned.

1, 1, 2, 4, 6, 10, 15, 23, 36, 59, 93, 148, 234, 372, 592, 939, 1490, 2366, 3754, 5959, 9460, 15017, 23838, 37841, 60068, 95352, 151362

Offset: 1

Author

Benedict W. J. Irwin, Dec 12 2016

Keywords

• nonn
• base

Comments

What combinatorial problem is this the answer to? - N. J. A. Sloane, Dec 21 2016

Appears to be strictly increasing for n > 1. - Chai Wah Wu, Dec 22 2016

Examples

			Left-align the binary representation of the Catalan numbers:
1
1
10
101
1110
101010
10000100
110101101
10110010110
The first column is all 1's, by definition.
The next column appears to have blocks of 1's and 0's.
The sizes of the blocks make this sequence.
The values of this column may be generated with the MMA code:
Table[If[2^Floor[Log2[CatalanNumber[n]]] + 2^(Floor[Log2[CatalanNumber[n]]] - 1) < CatalanNumber[n], 1, 0], {n, 1, 1000}].
		

Crossrefs

Cf. A000108.

Programs

Mathematica

Num = 10000;
T = Table[If[2^Floor[Log2[CatalanNumber[n]]] +2^(Floor[Log2[CatalanNumber[n]]] - 1) < CatalanNumber[n], 1, 0], {n, 1, Num}]; S = {};
For[i = 1, i < Num, i++, j = 0; If[T[[i]] == 1, While[T[[i]] == 1, i++; j++]; AppendTo[S, j];];]; S

Extensions

a(21)-a(27) from Chai Wah Wu, Dec 22 2016