

A172074


Continued fraction expansion of sqrt(12500)+50 = 100*phi, where phi=(sqrt(5)+1)/2 is the golden ratio.


1



161, 1, 4, 11, 1, 1, 3, 6, 1, 13, 8, 1, 6, 1, 4, 1, 1, 2, 1, 1, 1, 1, 13, 2, 1, 3, 8, 1, 2, 19, 1, 54, 1, 19, 2, 1, 8, 3, 1, 2, 13, 1, 1, 1, 1, 2, 1, 1, 4, 1, 6, 1, 8, 13, 1, 6, 3, 1, 1, 11, 4, 1, 222
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OFFSET

0,1


COMMENTS

The 62 trailing terms are repeated infinitely.
This is just one of an infinite set of continued fractions, related to the golden ratio, and more specifically to the square root of 125, 12500, 1250000...
Taking phi*10^k, one can look at sqrt(125) + 5, sqrt(12500) + 50 (this sequence, sqrt(1250000) + 500, etc.
Periodic with a period of length 62, starting right after the initial term. Moreover, the sequence is symmetric when any 54 or 222 is taken as central value (cf. formula).  M. F. Hasler, Sep 09 2011


LINKS



FORMULA

a(31*k  n) = a(31*k + n), for all n < 31k, k > 0.  M. F. Hasler, Sep 09 2011


MATHEMATICA

ContinuedFraction[N[Sqrt[12500], 50000], 63]
ContinuedFraction[100*GoldenRatio, 100] (* Harvey P. Dale, Dec 30 2018 *)


PROG

(PARI) default(realprecision, 199); contfrac((sqrt(5)+1)/.02) \\ M. F. Hasler, Sep 09 2011
(PARI) a(n)=[22261*!n, 1, 4, 11, 1, 1, 3, 6, 1, 13, 8, 1, 6, 1, 4, 1, 1, 2, 1, 1, 1, 1, 13, 2, 1, 3, 8, 1, 2, 19, 1, 54][32abs(n%6231)] \\ M. F. Hasler, Sep 09 2011


CROSSREFS



KEYWORD

cofr,nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



