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A280294 a(n) = a(n-1) + 2^n * a(n-2) with a(0) = 1 and a(1) = 1.

%I A280294 #29 Feb 16 2025 08:33:38
%S A280294 1,1,5,13,93,509,6461,71613,1725629,38391485,1805435581,80431196861,
%T A280294 7475495336637,666367860021949,123144883455482557,
%U A280294 21958686920654707389,8092381769059159562941,2886261393833112966453949,2124255587862077437434059453
%N A280294 a(n) = a(n-1) + 2^n * a(n-2) with a(0) = 1 and a(1) = 1.
%C A280294 The Rogers-Ramanujan continued fraction is defined by R(q) = q^(1/5)/(1+q/(1+q^2/(1+q^3/(1+ ... )))). The limit of a(n)/A015459(n+2) is 2^(-1/5) * R(2).
%H A280294 Seiichi Manyama, <a href="/A280294/b280294.txt">Table of n, a(n) for n = 0..114</a>
%H A280294 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%e A280294 1/1 = a(0)/A015459(2).
%e A280294 1/(1+2/1) = 1/3 = a(1)/A015459(3).
%e A280294 1/(1+2/(1+2^2/1)) = 5/7 = a(2)/A015459(4).
%e A280294 1/(1+2/(1+2^2/(1+2^3/1))) = 13/31 = a(3)/A015459(5).
%t A280294 nxt[{n_,a_,b_}]:={n+1,b,b+2^(n+1)*a}; NestList[nxt,{1,1,1},20][[All,2]] (* _Harvey P. Dale_, Jul 17 2020 *)
%o A280294 (Python)
%o A280294 def a():
%o A280294     a, b, p = 1, 0, 1
%o A280294     while True:
%o A280294         p, a, b = p + p, b, b + p * a
%o A280294         yield b
%o A280294 A280294 = a()
%o A280294 print([next(A280294) for _ in range(19)]) # _Peter Luschny_, Dec 05 2017
%Y A280294 Cf. A015463, A015459, A128915.
%Y A280294 Cf. similar sequences with the recurrence a(n-1) + q^n * a(n-2) for n>1, a(0)=1 and a(1)=1: this sequence (q=2), A279543 (q=3), A280340 (q=10).
%K A280294 nonn
%O A280294 0,3
%A A280294 _Seiichi Manyama_, Dec 31 2016