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Size of blocks of 1's in the second column of Catalan numbers written in binary and left-aligned.
2

%I #26 Dec 22 2016 22:12:25

%S 1,1,2,4,6,10,15,23,36,59,93,148,234,372,592,939,1490,2366,3754,5959,

%T 9460,15017,23838,37841,60068,95352,151362

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

%C What combinatorial problem is this the answer to? - _N. J. A. Sloane_, Dec 21 2016

%C Appears to be strictly increasing for n > 1. - _Chai Wah Wu_, Dec 22 2016

%e Left-align the binary representation of the Catalan numbers:

%e 1

%e 1

%e 10

%e 101

%e 1110

%e 101010

%e 10000100

%e 110101101

%e 10110010110

%e The first column is all 1's, by definition.

%e The next column appears to have blocks of 1's and 0's.

%e The sizes of the blocks make this sequence.

%e The values of this column may be generated with the MMA code:

%e Table[If[2^Floor[Log2[CatalanNumber[n]]] + 2^(Floor[Log2[CatalanNumber[n]]] - 1) < CatalanNumber[n], 1, 0], {n, 1, 1000}].

%t Num = 10000;

%t T = Table[If[2^Floor[Log2[CatalanNumber[n]]] +2^(Floor[Log2[CatalanNumber[n]]] - 1) < CatalanNumber[n], 1, 0], {n, 1, Num}]; S = {};

%t For[i = 1, i < Num, i++, j = 0; If[T[[i]] == 1, While[T[[i]] == 1, i++; j++]; AppendTo[S, j];];]; S

%Y Cf. A000108.

%K nonn,base

%O 1,3

%A _Benedict W. J. Irwin_, Dec 12 2016

%E a(21)-a(27) from _Chai Wah Wu_, Dec 22 2016