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

Offset

0,2

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

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A362583 A084188 A266721 * A033120 A365243 A091617

Adjacent sequences: A044429 A044430 A044431 * A044433 A044434 A044435

Keyword

nonn,base

Author

Clark Kimberling

Extensions

Offset fixed by Reinhard Zumkeller, Apr 07 2012

Status

approved