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A048678 Binary expansion of nonnegative integers expanded to "Zeckendorffian format" with rewrite rules 0->0, 1->01. 11
0, 1, 2, 5, 4, 9, 10, 21, 8, 17, 18, 37, 20, 41, 42, 85, 16, 33, 34, 69, 36, 73, 74, 149, 40, 81, 82, 165, 84, 169, 170, 341, 32, 65, 66, 133, 68, 137, 138, 277, 72, 145, 146, 293, 148, 297, 298, 597, 80, 161, 162, 325, 164, 329, 330, 661, 168, 337, 338, 677, 340 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

No two adjacent 1-bits. Permutation of A003714.

Replace 1 with 01 in binary. - Ralf Stephan, Oct 07 2003

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Transforms

FORMULA

a(n) = rewrite_0to0_1to01(n) [ Each 0->1, 1->10 in binary expansion of n ].

a(0)=0; a(n) = (3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2. - Benoit Cloitre, Aug 31 2003

a(0)=0, a(2n) = 2a(n), a(2n+1) = 4a(n) + 1. - Ralf Stephan, Oct 07 2003

EXAMPLE

11=1011 in binary, thus is rewritten as 100101 = 37 in decimal.

MAPLE

rewrite_0to0_1to01 := proc(n) option remember; if(n < 2) then RETURN(n); else RETURN(((2^(1+(n mod 2))) * rewrite_0to0_1to01(floor(n/2))) + (n mod 2)); fi; end;

MATHEMATICA

f[n_] := FromDigits[ Flatten[IntegerDigits[n, 2] /. {1 -> {0, 1}}], 2]; Table[f@n, {n, 0, 60}] (* Robert G. Wilson v, Dec 11 2009 *)

PROG

(PARI) a(n)=if(n<1, 0, (3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2)

(Haskell)

a048678 0 = 0

a048678 x = 2 * (b + 1) * a048678 x' + b

            where (x', b) = divMod x 2

-- Reinhard Zumkeller, Mar 31 2015

(Python)

from sympy import floor

def a(n): return 0 if n==0 else (3 - (-1)**n)*a(floor(n/2)) + (1 - (-1)**n)/2

print [a(n) for n in xrange(101)] # Indranil Ghosh, Jun 30 2017

CROSSREFS

Cf. A005203, A048679, A048680.

MASKTRANS transform of A053644.

Cf. A084471, A088697, A088698.

Cf. A124108.

Sequence in context: A227368 A120119 A298011 * A271586 A278508 A296208

Adjacent sequences:  A048675 A048676 A048677 * A048679 A048680 A048681

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen

STATUS

approved

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Last modified June 17 07:00 EDT 2019. Contains 324183 sequences. (Running on oeis4.)