A268726 - OEIS

%I #23 May 07 2021 08:44:25

%S 0,1,2,0,3,0,0,1,4,0,0,1,0,1,2,0,5,0,0,1,0,1,2,0,0,1,2,0,3,0,0,1,6,0,

%T 0,1,0,1,2,0,0,1,2,0,3,0,0,1,0,1,2,0,3,0,0,1,4,0,0,1,0,1,2,0,7,0,0,1,

%U 0,1,2,0,0,1,2,0,3,0,0,1,0,1,2,0,3,0,0,1,4,0,0,1,0,1,2,0,0,1,2,0,3,0,0,1,4,0,0,1,0,1,2,0,5,0,0,1,0,1,2,0,0

%N Index of the toggled bit between n and A268717(n+1): a(n) = A000523(A003987(n, A268717(1+n))).

%C A fractal sequence, because a permutation of A007814. Removing zeros yields A268727(n) = a(n)+1.

%H Antti Karttunen, <a href="/A268726/b268726.txt">Table of n, a(n) for n = 0..16383</a>

%H Indranil Ghosh, <a href="/A268726/a268726.txt">C program to generate the sequence</a>

%F a(n) = A007814(1 + A006068(n)).

%F a(n) = A000523(A003987(n, A268717(1+n))).

%F a(n) = floor(log_2(n XOR A003188(1 + A006068(n)))).

%F Other identities:

%F For all n >= 1, a(A003188(n-1)) = A007814(n).

%t A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m=A006068[Floor[n/2]]}, 2m + Mod[Mod[n,2] + Mod[m, 2], 2]]]; A268717[n_]:=If[n<1, 0, A003188[1 + A006068[n - 1]]]; a[n_]:= Floor[Log[2, BitXor[n, A268717[n + 1]]]]; Table[a[n], {n, 0, 200}] (* _Indranil Ghosh_, Apr 02 2017 *)

%o (Scheme) (define (A268726 n) (A000523 (A003987bi n (A268717 (+ 1 n))))) ;; A003987bi implements the bitwise-XOR, defined in A003987.

%o (PARI) A003188(n) = bitxor(n, n\2);

%o A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});

%o A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));

%o for(n=0, 200, print1(logint(bitxor(n, A268717(n + 1)), 2),", "))  \\ _Indranil Ghosh_, Apr 02 2017

%o (Python)

%o def A003188(n): return n^(n//2)

%o def A006068(n):

%o     if n<2: return n

%o     else:

%o         m=A006068(n//2)

%o         return 2*m + (n%2 + m%2)%2

%o def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))

%o print([int(math.floor(math.log(n^A268717(n + 1), 2))) for n in range(201)]) # _Indranil Ghosh_, Apr 02 2017

%Y One less than A268727.

%Y Cf. A003188, A006068.

%Y Cf. A000523, A003987, A007814, A101080, A268717.

%Y Cf. also array A268833.

%K nonn,base

%O 0,3

%A _Antti Karttunen_, Feb 13 2016