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

1, 2, 5, 11, 22, 45, 90, 181, 363, 726, 1453, 2907, 5814, 11629, 23258, 46517, 93035, 186070, 372141, 744282, 1488565, 2977131, 5954262, 11908525, 23817051, 47634102, 95268205, 190536410, 381072821, 762145643, 1524291286, 3048582573, 6097165147, 12194330294

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

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

Programs

Haskell

a044432 n = a044432_list !! n
a044432_list = scanl1 (\v b -> 2 * v + b) a005614_list
-- Reinhard Zumkeller, Apr 07 2012

Mathematica

FromDigits[(Floor[GoldenRatio(#+1)]-Floor[GoldenRatio #]-1)&@Range@#,2]&/@Range@40 (* Federico Provvedi, Oct 19 2018 *)
Floor[2^#/FromContinuedFraction[2^Fibonacci[Range[0,3*Max[1,Floor[2+Log[(#+1)/11]/ArcSinh[2]]]]]]]&/@Range[200] (* Federico Provvedi, Nov 01 2018 *)

Formula

a(n) = A000225(n+1) - A182028(n). - Reinhard Zumkeller, Apr 07 2012

a(n) = 2*a(n-1) + A005614(n) for n > 0, a(0) = 1. - Reinhard Zumkeller, Apr 07 2012

From Federico Provvedi, Oct 24 2018: (Start)

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

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)))).

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

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)

Extensions

Offset fixed by Reinhard Zumkeller, Apr 07 2012