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

1,2

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.

Links

Table of n, a(n) for n=1..60.

Example

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

Prog

(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:

Crossrefs

Cf. A086378, A087948.

Sequence in context: A047432 A095279 A030751 * A053427 A272054 A037354

Adjacent sequences: A087944 A087945 A087946 * A087948 A087949 A087950

Keyword

nonn

Author

Thomas Baruchel, Sep 07 2003

Status

approved