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%I #10 Sep 30 2015 08:28:40
%S 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,
%T 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,
%U 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
%N Minimum number of 12's such that n*[n; 12, ..., 12, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.
%C Sequence A261311 lists fixed points of this sequence.
%C 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).
%t 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 *)
%o (PARI) cf(v)={t=v[#v];forstep(i=#v-1,1,-1,t=v[i]+1/t);t}
%o 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))
%Y Cf. A213648, A262212 - A262220, A213900; A000057, A213891 - A213899.
%K nonn
%O 2,4
%A _M. F. Hasler_, Sep 15 2015
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