A092053 - OEIS

%I #18 Aug 14 2021 15:33:52

%S 1,1,7,19,53,81,823,5359,12923,21877,102061,354883,808865,1433689,

%T 25699639,369784999,817787423,1487830821,6512750579,23917578595,

%U 51908057021,96040578001,827937066989,6166467806391,13211837015707

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

%C 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).

%H Robert Israel, <a href="/A092053/b092053.txt">Table of n, a(n) for n = 1..1663</a>

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

%p R:= gfun:-rectoproc({r(n) = (r(n - 1))/(n - 1) + r(n - 2), r(1) = 0, r(2) = 1}, r(n), remember):

%p seq(numer(R(n)),n=2..30); # _Robert Israel_, May 14 2017

%t 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 *)

%t Table[Numerator[ContinuedFractionK[1, 1/k , {k, 1, n}]], {n, 1, 30}] (* _Vaclav Kotesovec_, Aug 14 2021 *)

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

%Y Cf. A001902, A092054.

%K frac,nonn

%O 1,3

%A _Paul D. Hanna_, Feb 19 2004