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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).

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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 *)

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Minimum number of 1112's such that n*[n; 11, 12, ..., 11, 12, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

1, 1, 1, 2, 1, 5, 3, 5, 4, 11, 7, 5, 11, 14, 9, 1, 11, 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, 11, 8, 13, 11, 19, 5, 43, 9, 5, 23, 45, 7, 5, 29, 17, 13, 12, 17, 29, 7, 5, 23, 11, 24, 19, 5, 59, 5, 30, 31, 5, 31, 20, 35, 9, 65, 17, 23, 5, 13, 59, 5, 23, 3, 8, 39, 11, 18, 17, 27, 3, 73, 29, 19, 29, 3, 13, 79, 23, 53, 19, 81, 5, 8, 43, 5, 59, 23, 15, 11, 55, 74, 35, 41, 26, 35, 9, 23, 35, 19, 14, 5, 41, 57, 59, 2, 5, 23, 47, 34, 11, 67, 17, 23, 119, 13, 11, 36, 56, 99, 17, 7, 83, 77, 31, 45, 59, 107, 11, 83, 23, 44, 131, 27, 15, 48, 5, 10, 59, 55, 23, 11, 47, 8929

2,14

Sequence A213900 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 fixed points (A213891 - A213899 and A261311).

A262211(n, d=1112)=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))

Cf. A213891 - A213900; A213648, A262212 - A262220, A213900; A000057, A213891 - A213899.

allocated for M. F. Hasler

Minimum number of 11's such that n*[n; 11, ..., 11, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

2, 3, 5, 4, 11, 7, 5, 11, 14, 1, 11, 6, 23, 19, 11, 8, 11, 17, 29, 7, 5, 23, 11, 24, 20, 35, 23, 13, 59, 5, 23, 3, 8, 39, 11, 18, 17, 27, 29, 3, 23, 43, 5, 59, 23, 15, 11, 55, 74, 35, 41, 26, 35, 9, 23, 35, 41, 57, 59, 2, 5, 23, 47, 34, 11, 67, 17, 23, 119, 13, 11, 36, 56, 99, 17, 7, 83, 77, 59, 107, 11, 83, 23, 44, 131, 27, 5, 10, 59, 55, 23, 11, 47, 89

2,1

Sequence A213900 lists fixed points of this sequence.

(PARI) cf(v)={t=v[#v]; forstep(i=#v-1, 1, -1, t=v[i]+1/t); t}

A262211(n, d=11)=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))

Cf. A213891 - A213900; A262212 - A262220.

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M. F. Hasler, Sep 15 2015

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allocated for M. F. Hasler

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