

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(m1))/(n*Fib(m+1)+Fib(m)) and n*[n,1,1,...,1,n] = n^2 + 1 + (n^2n1)*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. 3464.


LINKS

Table of n, a(n) for n=2..67.
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^2n1)*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



