A083741 a(n) = L(n) + a(L(n)), where L(n) = n - 2^floor(log_2(n)) (A053645).

0, 0, 0, 1, 0, 1, 2, 4, 0, 1, 2, 4, 4, 6, 8, 11, 0, 1, 2, 4, 4, 6, 8, 11, 8, 10, 12, 15, 16, 19, 22, 26, 0, 1, 2, 4, 4, 6, 8, 11, 8, 10, 12, 15, 16, 19, 22, 26, 16, 18, 20, 23, 24, 27, 30, 34, 32, 35, 38, 42, 44, 48, 52, 57, 0, 1, 2, 4, 4, 6, 8, 11, 8, 10, 12, 15, 16, 19, 22, 26, 16, 18, 20

Offset

0,7

Comments

a(2^j)=0. Local extrema are a(2^j-1) = 2^j-j-1 (A000295).

Links

Table of n, a(n) for n=0..82.

Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98 (1992), pp. 163-197, ex. 24. (PS file on author's web page.)

Formula

a(0)=0, a(1)=0, a(2n)=2a(n), a(4n+1)=2a(n)+a(2n+1), a(4n+3)=-2a(n)+3a(2n+1)+1.

a(n) = Sum_{i=0..k} i*2^e[i] where the binary expansion of n is n = Sum_{i=0..k} 2^e[i] with decreasing exponents e[0] > ... > e[k] (A272011). - Kevin Ryde, Apr 16 2024

Mathematica

f[l_]:=Join[l, l-1+Range[Length[l]]]; Nest[f, {0}, 7] (* Ray Chandler, Jun 01 2010 *)

Prog

(PARI) a(n)=if(n<2, 0, if(n%2==0, 2*a(n/2), if(n%4==1, 2*a((n-1)/4)+a((n+1)/ 2), -2*a((n-3)/4)+3*a((n-3)/2+1)+1)))
(PARI) a(n) = my(v=binary(n), c=-1); for(i=1, #v, if(v[i], v[i]=c++)); fromdigits(v, 2); \\ Kevin Ryde, Apr 16 2024

Crossrefs

Cf. A000295, A272011, A298043.

Sequence in context: A074078 A309635 A130659 * A258761 A256245 A173004

Adjacent sequences: A083738 A083739 A083740 * A083742 A083743 A083744

Keyword

easy,nonn,changed

Author

Ralf Stephan, May 05 2003

Extensions

Extended by Ray Chandler, Mar 04 2010

Status

approved