

A087947


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


1



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
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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+afloor(a); until (a := simplify(1/(afloor(a)), sqrt)) = i end_repeat: return(simplify((1isqrt(n)/sqrt(n))*(1+b/(p1)+1/afloor(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



