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a(1) = 1, a(n+1) is the numerator of n-th partial fraction of the continued fraction [1; 4, 16, 64, ..., 4^n].
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%I #20 Dec 31 2016 03:13:07

%S 1,5,81,5189,1328465,1360353349,5572008645969,91291791015909445,

%T 5982898821590650033489,1568381028778351153394849861,

%U 1644566705638271237843748737881425,6897812711726991987001765057444407253061,115726093792191122162903443021235072225308939601

%N a(1) = 1, a(n+1) is the numerator of n-th partial fraction of the continued fraction [1; 4, 16, 64, ..., 4^n].

%H Seiichi Manyama, <a href="/A280220/b280220.txt">Table of n, a(n) for n = 1..58</a>

%F a(n) = 4^(n-1)*a(n-1) + a(n-2).

%e G.f. = x + 5*x^2 + 81*x^3 + 5189*x^4 + 1328465*x^5 + 1360353349*x^6 + ...

%e a(3) = 81, the numerator of 1 + 1/(4 + 1/16) = 81/65.

%t f[n_] := Numerator[ FromContinuedFraction[ Reverse[4^Range[0, n -1]] ]]; Array[f, 12] (* _Robert G. Wilson v_, Dec 29 2016 *)

%Y Denominators are in A015475.

%Y Cf. A061377, A280219.

%K nonn,frac

%O 1,2

%A _Seiichi Manyama_, Dec 29 2016