A378992 a(n) = A011371(n) - A048881(n); The exponent of the highest power of 2 dividing the n-th factorial minus the exponent of the highest power of 2 dividing n-th Catalan number.

0, 0, 0, 1, 2, 2, 2, 4, 6, 6, 6, 7, 8, 8, 8, 11, 14, 14, 14, 15, 16, 16, 16, 18, 20, 20, 20, 21, 22, 22, 22, 26, 30, 30, 30, 31, 32, 32, 32, 34, 36, 36, 36, 37, 38, 38, 38, 41, 44, 44, 44, 45, 46, 46, 46, 48, 50, 50, 50, 51, 52, 52, 52, 57, 62, 62, 62, 63, 64, 64, 64, 66, 68, 68, 68, 69, 70, 70, 70, 73, 76, 76, 76

Offset: 0

Author

Antti Karttunen, Dec 16 2024

Keywords

• nonn
• easy

Comments

Apparently, after the initial three 0's, only terms of A092054 occur, every other as a single copy, and every other in a batch of 3 duplicated terms.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20000
• Index entries for sequences related to binary expansion of n

Crossrefs

Partial sums of A050605.

Cf. A000108, A000120, A000142, A007814, A011371, A048881.

Cf. also A092054, A204988, A050603, A136480.

Programs

Mathematica

A378992[n_] := n - DigitCount[n, 2, 1] - DigitCount[n + 1, 2, 1] + 1;
Array[A378992, 100, 0] (* or *)
MapIndexed[#2[[1]] - # &, Total[Partition[DigitCount[Range[0, 100], 2, 1], 2, 1], {2}]] (* Paolo Xausa, Dec 28 2024 *)

PARI

A378992(n) = (1+(n-hammingweight(n)-hammingweight(1+n)));

Formula

a(n) = A007814(A000142(n)) - A007814(A000108(n)) = A011371(n) - A048881(n).

a(0) = 0; for n > 0, a(n) = A050605(n-1) + a(n-1), where A050605(n) = A007814(n+1)+A007814(n+2)-1.