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%I #5 Mar 30 2012 18:36:44
%S 2,3,8,19,46,65,176,769,1714,2483,6680,15843,38366,54209,146784,
%T 1228481,2603746,3832227,10268200,24368627,59005454,83374081,
%U 225753616,986388545,2198530706,3184919251,8568369208,20321657667,49211684542
%N Numerators of the convergents in the continued fraction expansion for twice the constant given by A100338, where the partial quotients equal A006519 (greatest power of 2 dividing n) interleaved with 2's.
%C The convergents for the continued fraction of x are given by A100340(n)/A100341(n) and the convergents for the continued fraction of 2*x are given by A100342(n)/A100343(n), where A100342(n)/A100343(n) = 2*A100340(n)/A100341(n) for all n.
%F a(1) = 2, a(2) = 3; a(2*n) = a(2*n-1)*A006519(n) + a(2*n-2) for n>1, a(2*n-1) = 2*a(2*n-2) + a(2*n-3) for n>1.
%e The constant is 2*x=2.707742256859764748777788168033216248454666833624237..
%e contfrac(2*x) = [2;1, 2,2, 2,1, 2,4, 2,1, 2,2, 2,1, 2,8,... 2, A006519(n),... ].
%o (PARI) {a(n)=if(n==1,2,if(n==2,3,if(n%2==1,2*a(n-1)+a(n-2), a(n-1)*2^valuation(n/2,2)+a(n-2))))}
%Y Cf. A100338, A006519, A100340, A100341, A100343.
%K cofr,nonn
%O 1,1
%A _Paul D. Hanna_, Nov 18 2004
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