A048678 - OEIS

%I #57 Apr 24 2024 22:16:38

%S 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,

%T 81,82,165,84,169,170,341,32,65,66,133,68,137,138,277,72,145,146,293,

%U 148,297,298,597,80,161,162,325,164,329,330,661,168,337,338,677,340

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

%C No two adjacent 1-bits. Permutation of A003714.

%C Replace 1 with 01 in binary. - _Ralf Stephan_, Oct 07 2003

%H Reinhard Zumkeller, <a href="/A048678/b048678.txt">Table of n, a(n) for n = 0..10000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

%F a(0)=0; a(n) = (3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2. - _Benoit Cloitre_, Aug 31 2003

%F a(0)=0, a(2n) = 2a(n), a(2n+1) = 4a(n) + 1. - _Ralf Stephan_, Oct 07 2003

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

%p 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;

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

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

%o (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

%o (Haskell)

%o a048678 0 = 0

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

%o             where (x', b) = divMod x 2

%o -- _Reinhard Zumkeller_, Mar 31 2015

%o (Python)

%o def a(n):

%o     return 0 if n==0 else (3 - (-1)**n)*a(n//2) + (1 - (-1)**n)//2

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 30 2017

%o (Python)

%o def A048678(n): return int(bin(n)[2:].replace('1','01'),2) # _Chai Wah Wu_, Mar 18 2024

%Y Cf. A003714, A005203, A048679, A048680.

%Y MASKTRANS transform of A053644.

%Y Cf. A084471, A088697, A088698.

%Y Cf. A124108.

%K nonn,easy

%O 0,3

%A _Antti Karttunen_