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 A213648 The minimum number of 1's in the relation n*[n,1,1,...,1,n] = [x,...,x] between simple continued fractions. 24
 2, 3, 5, 4, 11, 7, 5, 11, 14, 9, 11, 6, 23, 19, 11, 8, 11, 17, 29, 7, 29, 23, 11, 24, 20, 35, 23, 13, 59, 29, 23, 19, 8, 39, 11, 18, 17, 27, 29, 19, 23, 43, 29, 59, 23, 15, 11, 55, 74, 35, 41, 26, 35, 9, 23, 35, 41, 57, 59, 14, 29, 23, 47, 34, 59, 67 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Multiplying n by a simple continued fraction with an increasing number of 1's sandwiched between n generates fractions that have a leading term x in their continued fraction, where x is obviously > n^2. We increase the number of 1's until the first and the last term in the simple terminating continued fraction of n*[n,1,...,1,n] =[x,...,x] is the same, x, and set a(n) to the count of these 1's. Conjecture: the fixed points of this sequence are in A000057. We have [n,1,1,...,1,n] = n + (n*Fib(m)+Fib(m-1))/(n*Fib(m+1)+Fib(m)) and n*[n,1,1,...,1,n] = n^2 + 1 + (n^2-n-1)*Fib(m)/(n*Fib(m+1)+Fib(m)), where m is the number of 1's. - Max Alekseyev, Aug 09 2012 The analog sequence with 11 instead of 1, A213900, seems to have the same fixed points, while other variants (A262212 - A262220, A262211) have other fixed points (A213891 - A213899,  A261311). - M. F. Hasler, Sep 15 2015 REFERENCES A. Hurwitz, Über die Kettenbrüche, deren Teilnenner arithmetische Reihen bilden, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Jahrg XLI, 1896, Jubelband II, S. 34-64. LINKS Bill Gosper, Appendix 2 Continued Fraction Arithmetic FORMULA Conjecture: a(n)=A001177(n)-1. EXAMPLE 3* [3,1,1,1,3] = [10,1,10],so a(3)=3 4* [4,1,1,1,1,1,4] = [18,2,18],so a(4)=5 5* [5,1,1,1,1,5] = [28,28],so a(5)=4 6* [6,1,1,1,1,1,1,1,1,1,1,1,6] = [39,1,2,2,2,1,39], so a(6)=11 7* [7,1,1,1,1,1,1,1,7] = [53,3,53], so a(7)=7 MAPLE A213648 := proc(n)         local h, ins, c ;         for ins from 1 do                 c := [n, seq(1, i=1..ins), n] ;                 h := numtheory[cfrac](n*simpcf(c), quotients) ;                 if op(1, h) = op(-1, h) then                         return ins;                 end if;         end do: end proc: # R. J. Mathar, Jul 06 2012 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[1, #] & /@ Range[2, 67] (* Michael De Vlieger, Sep 16 2015 *) PROG (PARI) {a(n) = local(t, m=1); if( n<2, 0, while( t = contfracpnqn( concat( [n, vector(m, i, 1 ), n])), t = contfrac( n * t[1, 1] / t[2, 1]); if( t[1] < n^2 || t[#t] < n^2, m++, break)); m)} /* Michael Somos, Jun 17 2012 */ (PARI) {a(n) = local(t, m=0); if( n<2, 0, until(t[1]==t[#t], m++; t = contfrac(n^2 + 1 + (n^2-n-1)*fibonacci(m)/(n*fibonacci(m+1)+fibonacci(m))); ); m )} /* Max Alekseyev, Aug 09 2012 */ CROSSREFS Cf. A000057, A262212 - A262220. Sequence in context: A112060 A084933 A213900 * A302849 A193971 A258861 Adjacent sequences:  A213645 A213646 A213647 * A213649 A213650 A213651 KEYWORD nonn AUTHOR Art DuPre, Jun 17 2012 STATUS approved

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Last modified December 5 22:37 EST 2019. Contains 329782 sequences. (Running on oeis4.)