This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
A003285(A003285(3))=1, 3 is the smallest number requiring 2 iterations to reach a square number, so a(2)=3.
with(numtheory): A065129 := proc(n) local m: if(n=1)then return 0:fi: for m from n by n to 5916*n do if(frac(sqrt(m))<>0)then if(n*nops(cfrac(sqrt(m),'periodic','quotients')[2])=m)then return m: fi: fi: od: return 0: end: seq(A065129(n),n=1..10); # Nathaniel Johnston, May 10 2011
Do[k = 2; While[ k / Length[ Last[ ContinuedFraction[ Sqrt[k]]]] != n, k++ ]; Print[k], {n, 2, 10} ]
For n=3, m = 9,10,11,12,13,14,15, A003285(11) = A003285(12), so 3 is not in the sequence.
a(2) = 2 because continued fraction of (1+sqrt(2))/2 = 1, 4, 1, 4, 1, 4, 1, ... has period (1,4) length 2.
A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic','quotients') ; nops(%[2]) ; else 0 ; fi; end: seq(A146326(n),n=1..100) ; # R. J. Mathar, Sep 06 2009
Table[cf = ContinuedFraction[(1 + Sqrt[n])/2]; If[Head[cf[[-1]]] === List, Length[cf[[-1]]], 0], {n, 100}]
f[n_] := Length@ ContinuedFraction[(1 + Sqrt[n])/2][[-1]]; Array[f, 100] (* Robert G. Wilson v, Apr 11 2017 *)
with(numtheory): for i from 1 to 150 do cfr := cfrac(ithprime(i)^(1/2), 'periodic','quotients'); printf(`%d,`, nops(cfr[2])) od:
Table[p=Prime[n]; Length[Last[ContinuedFraction[Sqrt[p]]]],{n,100}] (* T. D. Noe, May 22 2007 *)
Length[ContinuedFraction[Sqrt[#]][[2]]]&/@Prime[Range[100]] (* Harvey P. Dale, Sep 28 2024 *)
a(4) = 8*a(3)+a(2) = 8*536+66 = 4354.
I:=[2,8]; [n le 2 select I[n] else 8*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016
LinearRecurrence[{8,1},{2,8},30] (* Harvey P. Dale, Sep 21 2014 *)
RecurrenceTable[{a[0] == 2, a[1] == 8, a[n] == 8 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
x='x+O('x^30); Vec(2*(1-4*x)/(1-8*x-x^2)) \\ G. C. Greubel, Nov 07 2018
a[n_] := Catch[For[m = 1, True, m++, If[Length[ Last[ ContinuedFraction[ Sqrt[m] ]]] == n, Print[m]; Throw[m] ]]]; Table[a[n], {n, 0, 54}](* Jean-François Alcover, May 15 2012 *)
Flatten[Table[Position[Table[{s=Sqrt[n]};If[IntegerQ[s],0,Length[ ContinuedFraction[s] [[2]]]], {n,2000}],i,{1},1],{i,0,60}]] (* Harvey P. Dale, Sep 15 2013 *)
r[x_, n_] := Reduce[y > 0 && (x^2 - n*y^2 == -1 || x^2 - n*y^2 == 1 ), y, Integers];
a[n_ /; IntegerQ[ Sqrt[n]]] = 1;
a[n_] := a[n] = (k = 1; While[ r[k, n] === False, k++]; k);
Table[ Print[ a[n] ]; a[n], {n, 1, 67}] (* Jean-François Alcover, Jan 30 2012 *)
nmax = 500;
nconv = 200; (* The number of convergents 'nconv' should be increased if the linear recurrence is not found for some terms. *)
a[n_] := a[n] = Module[{lr}, If[IntegerQ[Sqrt[n]], 1, lr = FindLinearRecurrence[Numerator[Convergents[Sqrt[n], nconv]]]; SelectFirst[lr, #>1&]/2]];
Table[Print[n, " ", a[n] ]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Feb 22 2021 *)
nonSquares = Select[Range[120], !IntegerQ[Sqrt[#]]&]; a[n_] := Length[ Last[ ContinuedFraction[ Sqrt[ nonSquares[[n]] ]]]]; Table[a[n], {n, 1, Length[nonSquares]}] (* Jean-François Alcover, May 27 2013 *)
from math import isqrt from sympy.ntheory.continued_fraction import continued_fraction_periodic def A013943(n): return len(continued_fraction_periodic(0,1,n+(k:=isqrt(n))+int(n>=k*(k+1)+1))[-1]) # Chai Wah Wu, Jul 20 2024
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