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%I #7 Mar 07 2016 11:58:29
%S 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,
%T 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,
%U 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
%N 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))
%F a(n) -> add2c( (n-((trinv(n)*(trinv(n)-1))/2)), (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) )
%p 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;
%t trinv[n_] := Floor[(1/2)*(Sqrt[8*n + 1] + 1)];
%t Sum2c[a_, b_] := Sum2c[a, b] = If[0 == a || 0 == b, Return[0], Return[ Sum2c[BitXor[a, b], 2*BitAnd[a, b]] + 1]];
%t a[n_] := Sum2c[n - (1/2)*trinv[n]*(trinv[n] - 1), (trinv[n] - 1)*(trinv[ n]/2 + 1) - n];
%t Table[a[n], {n, 0, 120}](* _Jean-François Alcover_, Mar 07 2016, adapted from Maple *)
%Y Cf. A050600, A050602, A003056, A048720 (for the Maple implementation of trinv and XORnos, ANDnos)
%K nonn,tabl
%O 0,12
%A _Antti Karttunen_, Jun 22 1999
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