A092054 -id:A092054 - cp's OEIS frontend

A092053 Denominators of the convergents of the continued fraction expansion [1;1/2,1/3,1/4,...,1/n,...].

1, 1, 7, 19, 53, 81, 823, 5359, 12923, 21877, 102061, 354883, 808865, 1433689, 25699639, 369784999, 817787423, 1487830821, 6512750579, 23917578595, 51908057021, 96040578001, 827937066989, 6166467806391, 13211837015707

Offset: 1

Paul D. Hanna, Feb 19 2004

Numerators of convergents are A001902 (successive denominators of Wallis's product approximation to Pi/2). Sum of numerators and denominators equals powers of 2: A001902(n) + a(n) = 2^A092054(n).

Robert Israel, Table of n, a(n) for n = 1..1663

Cf. A001902, A092054.

R:= gfun:-rectoproc({r(n) = (r(n - 1))/(n - 1) + r(n - 2), r(1) = 0, r(2) = 1}, r(n), remember):
seq(numer(R(n)),n=2..30); # Robert Israel, May 14 2017

Numerator[RecurrenceTable[{r[n] == (r[n - 1])/(n - 1) + r[n - 2], r[1] == 0, r[2] == 1}, r, {n, 2, 30}]] (* Terry D. Grant, May 07 2017, fixed by Vaclav Kotesovec, Aug 14 2021 *)
Table[Numerator[ContinuedFractionK[1, 1/k , {k, 1, n}]], {n, 1, 30}] (* Vaclav Kotesovec, Aug 14 2021 *)

a(n)=local(A);CF=contfracpnqn(vector(n,k,1/k));A=denominator(CF[1,1]/CF[2,1])

a(n) = 2^A092054(n) - A001902(n).

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.

Original entry on oeis.org

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

Antti Karttunen, Dec 16 2024

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.

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

Partial sums of A050605.
Cf. A000108, A000120, A000142, A007814, A011371, A048881.
Cf. also A092054, A204988, A050603, A136480.

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 *)

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

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.

cp's OEIS Frontend

A092053 Denominators of the convergents of the continued fraction expansion [1;1/2,1/3,1/4,...,1/n,...].

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