A087947 Sum of successive remainders in computing Euclidean algorithm for (1, 1/sqrt(-n)) has real and imaginary parts equal.

1, 4, 5, 6, 8, 9, 16, 17, 18, 20, 24, 25, 36, 37, 38, 39, 40, 42, 48, 49, 64, 65, 66, 68, 72, 78, 80, 81, 100, 101, 102, 104, 105, 110, 117, 120, 121, 144, 145, 146, 147, 148, 150, 152, 155, 156, 164, 168, 169, 196, 197, 198, 200, 203, 210, 220, 222, 224, 225, 256

Offset: 1

Author

Thomas Baruchel, Sep 07 2003

Keywords

• nonn

Comments

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.

For all a(n) in the sequence, the relation: (2k)^2 <= a(n) <= (2k+1)^2 is true.

Examples

			kappa(1/sqrt(-203)) = (1/2 + (1/2)i) - (1/29 + (1/29)i)*sqrt(203).
		

Crossrefs

Cf. A086378, A087948.

Programs

MuPAD

kappa_1_over_comp_sqrt := proc(n) local a,b,i,p; begin if (a := sqrt(-n)-isqrt(-n)) = 0 then return(0) end_if: a := simplify(1/a,sqrt); i := a := simplify(1/(a - floor(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((1-isqrt(n)/sqrt(n))*(1+b/(p-1)+1/a-floor(1/a)),sqrt)); end_proc: