A048678 Binary expansion of nonnegative integers expanded to "Zeckendorffian format" with rewrite rules 0->0, 1->01.

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

Offset: 0

Author

Antti Karttunen

Keywords

• nonn
• easy

Comments

No two adjacent 1-bits. Permutation of A003714.

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

Examples

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A003714, A005203, A048679, A048680.

MASKTRANS transform of A053644.

Cf. A084471, A088697, A088698.

Cf. A124108.

Programs

Haskell

a048678 0 = 0
a048678 x = 2 * (b + 1) * a048678 x' + b
            where (x', b) = divMod x 2
-- Reinhard Zumkeller, Mar 31 2015

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 *)

PARI

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

PARI

a(n) = if(n == 0, 0, my(A = -2); sum(i = 0, logint(n, 2), A++; if(bittest(n, i), 1 << (A++)))) \\ Mikhail Kurkov, Mar 14 2024

Python

def a(n):
    return 0 if n==0 else (3 - (-1)**n)*a(n//2) + (1 - (-1)**n)//2
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017

Python

def A048678(n): return int(bin(n)[2:].replace('1','01'),2) # Chai Wah Wu, Mar 18 2024

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