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A044432 a(n) is the number whose base-2 representation is d(0)d(1)...d(n), where d=A005614 (the infinite Fibonacci word).

%I A044432 #58 Apr 19 2025 03:23:20
%S A044432 1,2,5,11,22,45,90,181,363,726,1453,2907,5814,11629,23258,46517,93035,
%T A044432 186070,372141,744282,1488565,2977131,5954262,11908525,23817051,
%U A044432 47634102,95268205,190536410,381072821,762145643,1524291286,3048582573,6097165147,12194330294
%N A044432 a(n) is the number whose base-2 representation is d(0)d(1)...d(n), where d=A005614 (the infinite Fibonacci word).
%C A044432 a(n) can also be calculated as floor(2^n * R), where the rabbit constant R=0.709803442861291314641787399444575597012... converges rapidly using the result from Davison described in the comments at A014565. - _Federico Provvedi_, Oct 24 2018
%H A044432 Reinhard Zumkeller, <a href="/A044432/b044432.txt">Table of n, a(n) for n = 0..1000</a>
%F A044432 a(n) = A000225(n+1) - A182028(n). - _Reinhard Zumkeller_, Apr 07 2012
%F A044432 a(n) = 2*a(n-1) + A005614(n) for n > 0, a(0) = 1. - _Reinhard Zumkeller_, Apr 07 2012
%F A044432 From _Federico Provvedi_, Oct 24 2018: (Start)
%F A044432 a(n) = A000079(n) * Sum_{k=0..n} ((floor(phi*(k+1)) - floor(phi*k) - 1)/2^k).
%F A044432 a(n) = floor(2^n*(1-Sum_{n >= 1} (-1)^(n+1)*(1+2^Fibonacci(3*n+1))/((2^(Fibonacci(3*n-1))-1)*(2^(Fibonacci(3*n + 2))-1)))).
%F A044432 a(n) = floor(2^n*R), where R is the rabbit constant.
%F A044432 a(n) = floor(2^n/[1, 2, 2, 4, 8, 32, ..., 2^Fibonacci(3*h)]), with h=1 for n=0, h=floor(2+log((n+1)/11)/arcsinh(2)) for n>0. (End)
%t A044432 FromDigits[(Floor[GoldenRatio(#+1)]-Floor[GoldenRatio #]-1)&@Range@#,2]&/@Range@40 (* _Federico Provvedi_, Oct 19 2018 *)
%t A044432 Floor[2^#/FromContinuedFraction[2^Fibonacci[Range[0,3*Max[1,Floor[2+Log[(#+1)/11]/ArcSinh[2]]]]]]]&/@Range[200] (* _Federico Provvedi_, Nov 01 2018 *)
%o A044432 (Haskell)
%o A044432 a044432 n = a044432_list !! n
%o A044432 a044432_list = scanl1 (\v b -> 2 * v + b) a005614_list
%o A044432 -- _Reinhard Zumkeller_, Apr 07 2012
%Y A044432 Cf. A000225, A005614, A182028, A000079, A014565.
%K A044432 nonn,base
%O A044432 0,2
%A A044432 _Clark Kimberling_
%E A044432 Offset fixed by _Reinhard Zumkeller_, Apr 07 2012