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
Links
- 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.)
Programs
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Mathematica
f[l_]:=Join[l,l-1+Range[Length[l]]]; Nest[f,{0},7] (* Ray Chandler, Jun 01 2010 *) -
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)))
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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
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
Extensions
Extended by Ray Chandler, Mar 04 2010
Comments