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A279026
Size of blocks of 1's in the second column of Catalan numbers written in binary and left-aligned.
2
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
1,3
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
EXAMPLE
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}].
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
CROSSREFS
Cf. A000108.
Sequence in context: A309173 A116084 A108925 * A120549 A167270 A355108
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
a(21)-a(27) from Chai Wah Wu, Dec 22 2016
STATUS
approved