A279026 -id:A279026 - cp's OEIS frontend

A279205 Length of second run of 1's in binary representation of Catalan(n).

0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 3, 4, 1, 3, 2, 1, 6, 1, 2, 1, 4, 7, 5, 2, 3, 1, 4, 2, 1, 1, 5, 2, 1, 3, 1, 1, 3, 3, 3, 3, 8, 2, 1, 2, 2, 1, 3, 2, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 4, 1, 2, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1

Offset: 0

N. J. A. Sloane, Dec 21 2016

Suggested by A279026.

What combinatorial problem is this the answer to?

			A000108(13) = 742900_10 = A264663(13) = 10110101010111110100_2, so a(13) = 2.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000108, A264663, A279026, A279206.

Q = {};
Num = 100;
T = Table[IntegerDigits[CatalanNumber[n], 2], {n, 0, Num}];
For[i = 1, i <= Num, i++,
c = 0; j = 1;
While[T[[i]][[j]] == 1, j++];
While[T[[i]][[j]] == 0, j++];
c = j;
While[T[[i]][[j]] == 1, j++];
c = j - c;
AppendTo[Q, c]
];
Q (* Benedict W. J. Irwin, Dec 21 2016 *)
Join[{0,0,0,1,0},Length[Split[IntegerDigits[#,2]][[3]]]&/@ CatalanNumber[ Range[5,100]]] (* Harvey P. Dale, Aug 20 2021 *)

a(19) to a(99) from Benedict W. J. Irwin, Dec 21 2016

A279206 Length of first run of 0's in binary representation of Catalan(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 4, 1, 1, 2, 5, 2, 2, 1, 1, 2, 4, 1, 3, 1, 4, 1, 1, 2, 2, 3, 4, 2, 1, 3, 1, 2, 3, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 3, 1, 1, 2, 8, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 6, 1, 3, 2, 1, 1, 2, 6, 1, 1, 1, 2, 2, 2, 3, 6, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Dec 22 2016

What combinatorial problem is this the answer to?

			A000108(13) = 742900_10 = A264663(13) = 10110101010111110100_2, so a(13) = 1.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000108, A264663, A279026, A279205.

f:= proc(n) local L; uses ListTools;
    L:= [1,op(convert(binomial(2*n,n)/(n+1),base,2))];
    L:= Reverse(L[2..-1]-L[1..-2]);
    Search(-1,L) - Search(1,L);
end proc:
map(f, [$0..100]); # Robert Israel, Dec 22 2016

Table[First[Map[Length, DeleteCases[Split@ IntegerDigits[CatalanNumber@ n, 2], w_ /; Times @@ w > 0]] /. {} -> {0}], {n, 0, 89}] (* Michael De Vlieger, Dec 22 2016 *)

cp's OEIS Frontend

A279205 Length of second run of 1's in binary representation of Catalan(n).

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