A071296 -id:A071296 - cp's OEIS frontend

A060215 Least a(n) such that the period of continued fraction for sqrt(a(n)) has at least n successive 1's.

2, 3, 7, 7, 13, 58, 58, 135, 461, 819, 2081, 13624, 13834, 35955, 95773, 244647, 639389, 1798800, 4374866, 11448871, 30002701, 78439683, 205337953, 541653136, 1407271538

Offset: 0

Jeff Burch, Mar 21 2001

a(25) <= 3684200835. - Chai Wah Wu, Jul 20 2024

			n  a(n)      C.f. period
0  2          1    2
1  3          2    1,2
2  7          4    1,1,1,4
3  7          4    1,1,1,4
4  13         5    1,1,1,1,6
5  58         7    1,1,1,1,1,1,14
6  58         7    1,1,1,1,1,1,14
7  135        8    1,1,1,1,1,1,1,22
8  461        15   2,8,10,1,1,1,1,1,1,1,1,10,8,2,42
9  819        10   1,1,1,1,1,1,1,1,1,56
10 2081       11   1,1,1,1,1,1,1,1,1,1,90
11 13624      62   1,2,1,...,4,5,1,1,1,1,1,1,1,1,1,1,1,5,4,...,1,2,1,232
12 13834      13   1,1,1,1,1,1,1,1,1,1,1,1,234
13 35955      14   1,1,1,1,1,1,1,1,1,1,1,1,1,378
14 95773      25   2,8,2,8,154,1,1,1,1,1,1,1,1,1,1,1,1,1,1,154,8,2,8,2,618
15 244647     16   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,988
16 639389     17   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1598
17 1798800    160  5,5,1,19,...,2,60,(17 1's),60,2,...,19,1,5,5,2682
18 4374866    19   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4182
19 11448871   20   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6766
20 30002701   35   2,8,2,8,2,8,2738,(20 1's),738,8,2,8,2,8,2,10954
21 78439683   22   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,17712
22 205337953  23   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,28658
23 541653136  1442 2,3,1,...,4,581,(23 1's),581,4,...,1,3,2,46546
24 1407271538 25   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,75026

Cf. A071296.

Table[Block[{k = 2}, While[NoneTrue[Map[Length, Select[If[IntegerQ@ #, {{#}}, Split@ Last@ ContinuedFraction@ #] &@ Sqrt@ k, First@ # == 1 &]], # >= n &], k++]; k], {n, 14}] (* Michael De Vlieger, Oct 26 2017 *)

print1("2, 3, 7, ");n=3;for(k=1,10^6,v=contfrac(sqrt(k));s=0; for(l=1,length(v)-n,if(v[l]==1,s=s+1,s=0);if(s==n,print1(k", ");n=n+1;k=k-1;break)))

More terms from Ralf Stephan, Mar 26 2003
Definition clarified by T. D. Noe, Apr 07 2014
a(11), a(14) corrected and a(17)-a(23) added by Lars Blomberg, Oct 24 2015
a(0), a(24) from Chai Wah Wu, Sep 23 2019

A309666 a(n) is the least k such that the denominators of continued fraction convergents for sqrt(k) match the first n Fibonacci numbers.

Original entry on oeis.org

2, 3, 7, 7, 13, 58, 58, 135, 819, 819, 2081, 13834, 13834, 35955, 244647, 244647, 639389, 4374866, 4374866, 11448871, 78439683, 78439683, 205337953, 1407271538, 1407271538, 3684200835, 25251313255, 25251313255, 66108441037, 453111560266, 453111560266, 1186259960295, 8130736409715, 8130736409715, 21286537898177

Offset: 1

Greg Dresden, Aug 11 2019

nonn

Aside from the first term, this appears to be a subset of A060215.
Same as A071296 if you drop a(0) and replace each repeated pair x,x with 0,x (credit to Daniel Suteu for pointing this out).
These are also the least a(n) such that the continued fraction expansion for sqrt(a(n) - floor(a(n))) begins with (n-1) 1's.

			For n = 5 the convergents of sqrt(13) are 3/1, 4/1, 7/2, 11/3, 18/5, 119/33, ... and the first five denominators are 1, 1, 2, 3, 5, which match the first five Fibonacci numbers. Since 13 is the first number with this property, then a(5) = 13.

Cf. A060215, A071296.

c = 1;
n = 2;
F = Table[Fibonacci[n], {n, 20}];
While[c <= 14,
If[! IntegerQ[Sqrt[n]]
   &&
   Denominator[Convergents[Sqrt[n], c]] == F[[1 ;; c]],
  Print[n, "  ", Denominator[Convergents[Sqrt[n], c]]];
  c++; n--];
n++
]

Conjectures from Colin Barker, Aug 26 2019: (Start)
G.f.: x*(2 + x + 4*x^2 - 42*x^3 - 15*x^4 - 39*x^5 + 100*x^6 + x^7 - 61*x^8 + 172*x^9 + 31*x^10 - 17*x^11 + 26*x^12 - 2*x^13 + x^14 - 2*x^15) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 - x + x^2)*(1 - x - x^2)*(1 + x + 2*x^2 - x^3 + x^4)*(1 + 3*x + 8*x^2 + 3*x^3 + x^4)).
a(n) = a(n-1) + 21*a(n-3) - 21*a(n-4) - 50*a(n-6) + 50*a(n-7) - 86*a(n-9) + 86*a(n-10) - 13*a(n-12) + 13*a(n-13) + a(n-15) - a(n-16) for n>16.
(End)

cp's OEIS Frontend

A060215 Least a(n) such that the period of continued fraction for sqrt(a(n)) has at least n successive 1's.

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