login
a(n) is the number whose base-2 representation is d(0)d(1)...d(n), where d=A005614 (the infinite Fibonacci word).
4

%I #53 Dec 20 2018 23:39:18

%S 1,2,5,11,22,45,90,181,363,726,1453,2907,5814,11629,23258,46517,93035,

%T 186070,372141,744282,1488565,2977131,5954262,11908525,23817051,

%U 47634102,95268205,190536410,381072821

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

%C 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 Reinhard Zumkeller, <a href="/A044432/b044432.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A000225(n+1) - A182028(n). - _Reinhard Zumkeller_, Apr 07 2012

%F a(n) = 2*a(n-1) + A005614(n) for n > 0, a(0) = 1. - _Reinhard Zumkeller_, Apr 07 2012

%F From _Federico Provvedi_, Oct 24 2018: (Start)

%F a(n) = A000079(n) * Sum_{k=0..n} ((floor(phi*(k+1)) - floor(phi*k) - 1)/2^k).

%F 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 a(n) = floor(2^n*R), where R is the rabbit constant.

%F 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.

%F (End)

%t FromDigits[(Floor[GoldenRatio(#+1)]-Floor[GoldenRatio #]-1)&@Range@#,2]&/@Range@40 (* _Federico Provvedi_, Oct 19 2018 *)

%t 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 (Haskell)

%o a044432 n = a044432_list !! n

%o a044432_list = scanl1 (\v b -> 2 * v + b) a005614_list

%o -- _Reinhard Zumkeller_, Apr 07 2012

%Y Cf. A000225, A005614, A182028, A000079, A014565.

%K nonn,base

%O 0,2

%A _Clark Kimberling_

%E Offset fixed by _Reinhard Zumkeller_, Apr 07 2012