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A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)*f(n+1)+1, and f(2n+1) = f(n)*f(n+1)+1 for n > 2.

%I A338637 #47 Jan 04 2021 08:45:54
%S A338637 1,1,3,5,7,9,19,29,43,53,79,149,187,293,583,849,1311,1601,2343,3525,
%T A338637 4315,8025,12027,15029,28119,44169,55303,109533,171843,249781,495991,
%U A338637 766361,1115087,1361297,2103007,3075769,3755239,5651717,8267267,10118237
%N A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)*f(n+1)+1, and f(2n+1) = f(n)*f(n+1)+1 for n > 2.
%H A338637 Georg Fischer, <a href="/A338637/b338637.txt">Table of n, a(n) for n = 1..1000</a>
%e A338637 f(1) = 1/1, so a(1) = 1.
%e A338637 f(2) = 1/2, so a(2) = 1.
%e A338637 f(3) = f(1) * f(2) + 1 = 3/2, so a(3) = 3.
%e A338637 f(4) = f(1) * f(3) + 1 = 1 * 3/2 + 1 = 5/2, so a(4) = 5.
%e A338637 f(5) = f(2) * f(3) + 1 = 1/2 * 3/2 + 1 = 7/4, so a(5) = 7.
%e A338637 f(n) for n>=1: 1, 1/2, 3/2, 5/2, 7/4, 9/4, 19/4, 29/8, 43/8, 53/8, 79/16, 149/16, 187/16, 293/32, 583/32, 849/32 ...
%o A338637 (PARI) f(n) = if (n<=2, 1/n, my(x=n\2); if (n%2, f(x)*f(x+1)+1, f(x-1)*f(x+1)+1));
%o A338637 a(n) = numerator(f(n)); \\ _Michel Marcus_, Nov 05 2020
%Y A338637 Denominators are given in A173862.
%K A338637 nonn,frac
%O A338637 1,3
%A A338637 _William Phoenix Marcum_, Nov 04 2020