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

1,3

Comments

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).

Links

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

Formula

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

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A001902, A092054.

Sequence in context: A155414 A155374 A155272 * A072630 A002714 A126361

Adjacent sequences: A092050 A092051 A092052 * A092054 A092055 A092056

Keyword

frac,nonn

Author

Paul D. Hanna, Feb 19 2004

Status

approved