cp's OEIS Frontend

A213900 The minimum number of 11's in the relation n*[n,11,11,...,11,n] = [x,...,x] between simple terminating continued fractions.

%I A213900 #29 Sep 17 2015 04:40:12
%S A213900 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,
%T A213900 23,13,59,5,23,3,8,39,11,18,17,27,29,3,23,43,5,59,23,15,11,55,74,35,
%U A213900 41,26,35,9,23,35,41,57,59,2,5,23,47,34,11,67,17,23,119,13
%N A213900 The minimum number of 11's in the relation n*[n,11,11,...,11,n] = [x,...,x] between simple terminating continued fractions.
%C A213900 In a variant of A213891, multiply n by a number with simple continued fraction [n,11,11,..,11,n] and increase the number of 11's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
%C A213900 2 * [2, 11, 11, 2] = [4, 5, 1, 1, 5, 4],
%C A213900 3 * [3, 11, 11, 11, 3] = [9, 3, 1, 2, 3, 2, 1, 3, 9],
%C A213900 4 * [4, 11, 11, 11, 11, 11, 4] = [16, 2, 1, 3, 2, 1, 1, 10, 1, 1, 2, 3, 1, 2, 16],
%C A213900 5 * [5, 11, 11, 11, 11, 5] = [25, 2, 4, 1, 1, 2, 2, 1, 1, 4, 2, 25] ,
%C A213900 6 * [6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 6] = [36, 1, 1, 5, 1, 1, 2, 7, 16, 1, 1, 1, 2, 1, 6, 1, 2, 1, 1, 1, 16, 7, 2, 1, 1, 5, 1, 1, 36].
%C A213900 The number of 11's needed defines the sequence a(n).
%C A213900 If we consider the fixed points such that a(n)=n, we conjecture to obtain the sequence A000057. This sequence consists of prime numbers. We conjecture that this sequence of prime numbers, in addition to its well-known relation to the collection of Fibonacci sequences (sequences satisfying f(n)=f(n-1)+f(n-2) with arbitrary positive integer values for f(1) and f(2)) it also refers to the sequences satisfying f(n)=11*f(n-1)+f(n-2), A049666, A015457, etc. This would mean that a prime is in the sequence A000057 if and only if it divides some term in each of the sequences satisfying f(n)=11*f(n-1)+f(n-2).
%C A213900 It is surprising that the fixed points of this sequence seem to be the same as for the variant A213648 where 11 is replaced by 1, while for the other variants A262212 - A262220 (where the repeated term is 2, ..., 10) the fixed points are different, see A213891 - A213899. - _M. F. Hasler_, Sep 15 2015
%t A213900 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[11, #] & /@ Range[2, 120] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A213900 (PARI) \\ This PARI program will generate sequence A000057
%o A213900 {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213900    t = contfracpnqn( concat([n, vector(m,i,11), n]));
%o A213900    t = contfrac(n*t[1,1]/t[2,1]);
%o A213900    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213900 m)};
%o A213900 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213900 Cf. A000057, A213358, A213891 - A213899.
%Y A213900 Cf. A213648, A262212 - A262220, A213900.
%K A213900 nonn
%O A213900 2,1
%A A213900 _Art DuPre_, Jun 24 2012