

A262211


Minimum number of 12's such that n*[n; 12, ..., 12, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.


20



1, 1, 1, 2, 1, 5, 3, 5, 5, 9, 1, 6, 5, 5, 7, 8, 5, 19, 5, 5, 9, 23, 3, 14, 13, 17, 5, 2, 5, 31, 15, 9, 17, 5, 5, 36, 19, 13, 11, 19, 5, 43, 9, 5, 23, 45, 7, 5, 29, 17, 13, 12, 17, 29, 11, 19, 5, 59, 5, 30, 31, 5, 31, 20, 9, 65, 17, 23, 5, 13, 11, 3, 73, 29, 19, 29, 13, 79, 23, 53, 19, 81, 5, 8, 43, 5, 19, 14, 5, 41, 23, 31, 45, 59, 15, 48, 5, 29
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OFFSET

2,4


COMMENTS

Sequence A261311 lists fixed points of this sequence.
It is surprising that the variant A213900 with 11 instead of 12 has the same fixed points A000057 as the variant A213648 with 1 instead of 12, but other variants (A262212  A262220 and this one) have different sets of fixed points (A213891  A213899 and A261311).


LINKS

Table of n, a(n) for n=2..99.


MATHEMATICA

f[m_, n_] := Block[{c, k = 1}, c[x_, y_] := ContinuedFraction[x FromContinuedFraction[Join[{x}, Table[m, {y}], {x}]]]; While[First@ c[n, k] != Last@ c[n, k], k++]; k]; f[12, #] & /@ Range[2, 120] (* Michael De Vlieger, Sep 16 2015 *)


PROG

(PARI) cf(v)={t=v[#v]; forstep(i=#v1, 1, 1, t=v[i]+1/t); t}
A262211(n, d=12)=for(k=1, 9e9, (c=contfrac(cf(vector(k+2, i, if(i>1&&i<k+2, d, n)))*n))[1]==c[#c]&&return(k))


CROSSREFS

Cf. A213648, A262212  A262220, A213900; A000057, A213891  A213899.
Sequence in context: A076062 A295563 A132582 * A094512 A182650 A127367
Adjacent sequences: A262208 A262209 A262210 * A262212 A262213 A262214


KEYWORD

nonn


AUTHOR

M. F. Hasler, Sep 15 2015


STATUS

approved



