%N Sum of successive remainders in computing euclidean algorithm for (1, -1/sqrt(-n)) has real and imaginary parts equal.
%C Since the computation of the algorithm needs an extension of the integer part over a subset of C, the rule: floor(i*x) = i*floor(x) is used (which is what MuPAD does). The following program computes the exact value of the sum.
%e kappa(-1/sqrt(-105)) = -(1/210 + (1/210)i)*sqrt(105).
%o (MuPAD) kappa_neg_1_over_comp_sqrt := proc(n) local a,b,i,p; begin if (a := -sqrt(-n)+ceil(sqrt(-n))) = 0 then return(0) end_if: i := a := simplify(1/a,sqrt); p := 1; b := 0; repeat p := p*a; b := b*a+a-floor(a); until (a := simplify(1/(a-floor(a)),sqrt)) = i end_repeat: return(simplify(-(b/(p-1) + 1/a)/sqrt(-n),sqrt)); end_proc:
%Y Cf. A086378, A087947.
%A _Thomas Baruchel_, Sep 07 2003