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

0,12

Links

Table of n, a(n) for n=0..119.

Formula

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

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 *)

Crossrefs

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

Sequence in context: A108803 A348920 A348921 * A101650 A287411 A053796

Adjacent sequences: A050598 A050599 A050600 * A050602 A050603 A050604

Keyword

nonn,tabl

Author

Antti Karttunen, Jun 22 1999

Status

approved