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A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

%I A073415 #19 Jun 10 2020 07:52:33
%S A073415 1,1,5,11,49,207,1291,5371,12033,53503,333051,719605,3211471,19988431,
%T A073415 83165195,352649211,788463617,3506503679,21827485691,47161475061,
%U A073415 210473385935,889055018801,5544803498741,23068269013765,51681341526271
%N A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).
%H A073415 Robert Israel, <a href="/A073415/b073415.txt">Table of n, a(n) for n = 1..1650</a>
%F A073415 a(n) = a(n-1) A007400(n) + a(n-2). - _Robert Israel_, Jun 14 2016
%p A073415 a007400:= proc(n) option remember; local n8, n16;
%p A073415     n8:= n mod 8;
%p A073415     if n8 = 0 or n8 = 3 then return 2
%p A073415     elif n8 = 4 or n8 = 7 then return 4
%p A073415     elif n8 = 1 then return procname((n+1)/2)
%p A073415     elif n8 = 2 then return procname((n+2)/2)
%p A073415     fi;
%p A073415     n16:= n mod 16;
%p A073415     if n16 = 5 or n16 = 14 then return 4
%p A073415     elif n16 = 6 or n16 = 13 then return 6
%p A073415     fi
%p A073415 end proc:
%p A073415 a007400(0):= 0: a007400(1):= 1: a007400(2):= 4:
%p A073415 A[1]:= 1: A[2]:= 1:
%p A073415 for n from 3 to 100 do
%p A073415   A[n]:= A[n-1]*a007400(n-1)+A[n-2];
%p A073415 od:
%p A073415 seq(A[n], n=1..100); # _Robert Israel_, Jun 14 2016
%t A073415 (* b is a007400 *)
%t A073415 b[n_] := b[n] = Module[{n8, n16}, n8 = Mod[n, 8]; Which[n8 == 0 || n8 == 3, Return[2], n8 == 4 || n8 == 7, Return[4], n8 == 1, Return[b[(n+1)/2]], n8 == 2, Return[b[(n+2)/2]]]; n16 = Mod[n, 16]; Which[n16 == 5 || n16 == 14, Return[4], n16 == 6 || n16 == 13, Return[6]]];
%t A073415 b[0] = 0; b[1] = 1; b[2] = 4;
%t A073415 a[1] = a[2] = 1;
%t A073415 a[n_] := a[n] = a[n-1] b[n-1] + a[n-2];
%t A073415 Array[a, 100] (* _Jean-François Alcover_, Jun 10 2020, after _Robert Israel_ *)
%o A073415 (PARI) a(n)=component(component(contfracpnqn(contfrac(sum(k=0,20,1/2^(2^k)),n)),1),2)
%Y A073415 Cf. A007400, A007404, A073414.
%K A073415 easy,frac,nonn
%O A073415 1,3
%A A073415 _Benoit Cloitre_, Aug 23 2002