A050601 Recursion counts for summation table A003056 with formula a(0,x) = x, a(y,0) = y, a(y,x) = a((y XOR x),2*(y AND x)).

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 0, 0, 1, 2, 2, 1, 0, 0, 2, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 3, 2, 3, 1, 3, 2, 3, 0, 0, 1, 3, 3, 2, 2, 3, 3, 1, 0, 0, 2, 1, 3, 2, 1, 2, 3, 1, 2, 0, 0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 0, 3, 2, 3, 1, 3, 1, 3, 1, 3, 2, 3, 0, 0, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 0, 0, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 0

Offset: 0

Author

Antti Karttunen, Jun 22 1999

Keywords

• nonn
• tabl

Crossrefs

Cf. A050600, A050602, A003056, A048720 (for the Maple implementation of trinv and XORnos, ANDnos)

Programs

Maple

add2c := proc(a,b) option remember; if((0 = a) or (0 = b)) then RETURN(0); else RETURN(1+add_c(XORnos(a,b),2*ANDnos(a,b))); fi; end;

Mathematica

trinv[n_] := Floor[(1/2)*(Sqrt[8*n + 1] + 1)];
Sum2c[a_, b_] := Sum2c[a, b] = If[0 == a || 0 == b, Return[0], Return[ Sum2c[BitXor[a, b], 2*BitAnd[a, b]] + 1]];
a[n_] := Sum2c[n - (1/2)*trinv[n]*(trinv[n] - 1), (trinv[n] - 1)*(trinv[ n]/2 + 1) - n];
Table[a[n], {n, 0, 120}](* Jean-François Alcover, Mar 07 2016, adapted from Maple *)

Formula

a(n) -> add2c( (n-((trinv(n)*(trinv(n)-1))/2)), (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) )