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

0,5

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

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.

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 *)

Prog

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

Crossrefs

Partial sums of A050605.

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

Cf. also A092054, A204988, A050603, A136480.

Sequence in context: A361404 A248781 A236840 * A182539 A170887 A103265

Adjacent sequences: A378989 A378990 A378991 * A378994 A378995 A378996

Keyword

nonn,easy,new

Author

Antti Karttunen, Dec 16 2024

Status

approved