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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). 4
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 (list; graph; refs; listen; history; text; internal format)
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: A293362 A084188 A266721 * A033120 A091617 A205880

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

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Last modified August 13 02:52 EDT 2020. Contains 336441 sequences. (Running on oeis4.)