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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
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.
This is not an efficient way to calculate phi. - Franklin T. Adams-Watters, Sep 10 2011
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
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)=[222-61*!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][32-abs(n%62-31)] \\ M. F. Hasler, Sep 09 2011
CROSSREFS
Sequence in context: A091550 A027553 A159378 * A171223 A278896 A249397
KEYWORD
cofr,nonn,nice,easy
AUTHOR
Shane Findley, Jan 25 2010
EXTENSIONS
Clarified the definition, following an observation by Franklin T. Adams-Watters. M. F. Hasler, Sep 09 2011
STATUS
approved