A065938 -id:A065938 - cp's OEIS frontend

A003285 Period of continued fraction for square root of n (or 0 if n is a square).

0, 1, 2, 0, 1, 2, 4, 2, 0, 1, 2, 2, 5, 4, 2, 0, 1, 2, 6, 2, 6, 6, 4, 2, 0, 1, 2, 4, 5, 2, 8, 4, 4, 4, 2, 0, 1, 2, 2, 2, 3, 2, 10, 8, 6, 12, 4, 2, 0, 1, 2, 6, 5, 6, 4, 2, 6, 7, 6, 4, 11, 4, 2, 0, 1, 2, 10, 2, 8, 6, 8, 2, 7, 5, 4, 12, 6, 4, 4, 2, 0, 1, 2, 2, 5, 10, 2, 6, 5, 2, 8, 8, 10, 16, 4, 4, 11, 4, 2, 0, 1, 2, 12

Offset: 1

N. J. A. Sloane

Any string of five consecutive terms m^2 - 2 through m^2 + 2 for m > 2 in the sequence has the corresponding periods 4,2,0,1,2. - Lekraj Beedassy, Jul 17 2001
For m > 1, a(m^2+m) = 2 and the continued fraction is m, 2, 2*m, 2, 2*m, 2, 2*m, ... - Arran Fernandez, Aug 14 2011
Apparently the generating function of the sequence for the denominators of continued fraction convergents to sqrt(n) is always rational and of form p(x)/[1 - C*x^m + (-1)^m * x^(2m)], or equivalently, the denominators satisfy the linear recurrence b(n+2m) = C*b(n+m) - (-1)^m * b(n), where a(n) is equal to m for each nonsquare n, or 0. See A006702 for the conjecture regarding C. The same conjectures apply to the sequences of the numerators of continued fraction convergents to sqrt(n). - Ralf Stephan, Dec 12 2013
If a(n)=1, n is of form k^2+1 (A002522 except the initial term 1). See A013642 for a(n)=2, A013643 for a(n)=3, A013644 for a(n)=4, A010337 for a(n)=5, A020347 for a(n)=6, A010338 for a(n)=7, A020348 for a(n)=8, A010339 for a(n)=9, and furthermore A020349-A020439. - Ralf Stephan, Dec 12 2013
From William Krier, Dec 12 2024: (Start)
a(m^2-4) = 4 for even m>=6 since sqrt(m^2-4) = [m-1; 1, (m-4)/2, 1, 2*(m-1)].
a(m^2-4) = 6 for odd m>=5 since sqrt(m^2-4) = [m-1; 1, (m-3)/2, 2, (m-3)/2, 1, 2*(m-1)].
a(m^2+4) = 2 for even m>=2 since sqrt(m^2+4) = [m; m/2, 2*m].
a(m^2+4) = 5 for odd m>=3 since sqrt(m^2+4) = [m; (m-1)/2, 1, 1, (m-1)/2, 2*m]. (End)

A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 197.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from T. D. Noe)
Marius Beceanu, Period of the Continued Fraction of sqrt(n), (Feb 05 2003)
Leon Bernstein, Fundamental units and cycles in the period of real quadratic number fields, I. Pacific J. Math 63 (1976): 37-61.
Ron Knott, All square-root continued fractions eventually repeat
R. Luczak, Patterns in the period lengths of simple periodic continued fractional representations of square roots of integers near perfect squares, QED: Chicago's Youth Math Research Symposium (April 2013).
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Justin T. Miller, Families of Continued Fractions, translated (2000) from a document of Nikos Drakos, Computer Based Learning Unit, University of Leeds.
C. D. Patterson and H. C. Williams, Some Periodic Continued Fractions with Long Periods, Mathematics of Computation, Vol. 44 (1985), No. 170, pp. 523-532.
A. M. Rockett and P. Szuesz, On the lengths of the periods of the continued fractions of square-roots of integers, Forum Mathematicum, 2 (1990), 119-123.
R. G. Stanton, C. Sudler, Jr. and H. C. Williams, An Upper Bound for the Period of the Simple Continued Fraction for Sqrt(D), Pacific Journal of Math., Vol. 67 (1976), No. 2, pp. 525-536.
Hanna Uscka-Wehlou, Continued Fractions and Digital Lines with Irrational Slopes, in Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Volume 4992/2008, Springer-Verlag.
A. J. van der Poorten, Fractional Parts of the Period of the Continued Fraction Expansion of Quadratic Integers [Refined and revised text of a talk given at the 2nd Conference of the Canadian Number Theory Association, Vancouver, 1989]
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
H. C. Williams, A Numerical Investigation Into the Length of the Period of the Continued Fraction Expansion of Sqrt(D), Mathematics of Computation, Vol. 36 (1981), No. 154, pp. 593-601.

Cf. A013943, A035015, A054269, A061490, A065938, A067280, A097853.

f:= n ->  if issqr(n) then 0
   else nops(numtheory:-cfrac(sqrt(n),'periodic','quotients')[2]) fi:
map(f, [$1..100]); # Robert Israel, Sep 02 2015

a[n_] := ContinuedFraction[Sqrt[n]] // If[Length[ # ] == 1, 0, Length[Last[ # ]]]&
pcf[n_]:=Module[{s=Sqrt[n]},If[IntegerQ[s],0,Length[ContinuedFraction[s][[2]]]]]; Array[pcf,110] (* Harvey P. Dale, Jul 15 2017 *)

a(n)=if(issquare(n),return(0));my(s=sqrt(n),x=s,f=floor(s),P=[0],Q=[1],k);while(1,k=#P;P=concat(P,f*Q[k]-P[k]);Q=concat(Q,(n-P[k+1]^2)/Q[k]);k++;for(i=1,k-1,if(P[i]==P[k]&&Q[i]==Q[k],return(k-i)));x=(P[k]+s)/Q[k];f=floor(x)) \\ Charles R Greathouse IV, Jul 31 2011

isok(n, p) = {localprec(p); my(cf = contfrac(sqrt(n))); setsearch(Set(cf), 2*cf[1]);}
a(n) = {if (issquare(n), 0, my(p=100); while (! isok(n, p), p+=100); localprec(p); my(cf = contfrac(sqrt(n))); for (k=2, #cf, if (cf[k] == 2*cf[1], return (k-1))););} \\ Michel Marcus, Jul 07 2021

from sympy.ntheory.continued_fraction import continued_fraction_periodic
def a(n):
    cfp = continued_fraction_periodic(0, 1, d=n)
    return 0 if len(cfp) == 1 else len(cfp[1])
print([a(n) for n in range(1, 104)]) # Michael S. Branicky, Aug 22 2021

A065936 a(n) is the integer (reduced squarefree) under the square root obtained when the inverse of a variant of Minkowski's question mark function is applied to the n-th ratio A007305(n+1)/A007306(n+1) in the left-hand subtree of Stern-Brocot tree and zero when it results a rational value.

Original entry on oeis.org

0, 5, 5, 0, 2, 2, 0, 2, 3, 0, 3, 3, 0, 3, 2, 5, 13, 17, 2, 17, 37, 5, 13, 13, 5, 37, 17, 2, 17, 13, 5, 3, 17, 3, 37, 21, 13, 10, 37, 3, 401, 6, 13, 10, 401, 0, 17, 17, 0, 401, 10, 13, 6, 401, 3, 37, 10, 13, 21, 37, 3, 17, 3, 0, 37, 10, 0, 401, 506, 17, 5, 401, 37, 21610, 730, 5, 1373

Offset: 1

Antti Karttunen, Dec 07 2001

nonn

Note: the underlying function N2Qv (see the Maple code) maps natural numbers 1, 2, 3, 4, 5, ..., through all the positive rationals in the open range (0,1): 1/2, 1/3, 2/3, 1/4, 2/5, 3/5, ... bijectively to the union of positive rationals and quadratic surds. A065937 gives similar mapping involving the inverse of the standard Minkowski's question mark function.

Note the symmetry of rows 0; 5,5; 0,2,2,0; 2,3,0,3,3,0,3,2; 5,13,17,2,17,37,5,13,13,5,37,17,2,17,13,5; ... emanating from the symmetry present in A007306.

			The first few values for this mapping are N2Qv(1) = 1, N2Qv(2) = (sqrt(5)-1)/2, N2Qv(3) = (sqrt(5)+1)/2, N2Qv(4) = 1/2, N2Qv(5) = sqrt(2)/2, N2Qv(6) = sqrt(2), N2Qv(7) = 2, N2Qv(8) = sqrt(2)-1

Eric Weisstein's World of Mathematics, Quadratic Irrational Number.
Index entries for sequences related to Minkowski's question mark function
Index entries for sequences related to Stern's sequences

a(n) = A065937(A065934(n)). Positions of the zeros are given by A065810. Positions of sqrt(n) in this mapping: A065938.

[seq(find_sqrt(N2Qv(j)),j=1..512)];
N2Qv := proc(n) local m; m := n + 2^floor_log_2(n); Inverse_of_Variant_of_MinkowskisQMark(A007305(m+1)/A047679(m-1)); end;
Inverse_of_Variant_of_MinkowskisQMark := proc(r) local x,y,b,d,k,s,i,q; x := numer(r); y := denom(r); if(y = 2*x) then RETURN(1); fi; b := []; d := []; k := 0; s := 0; i := 0; while(x <> 0) do q := floor(x/y); if(i > 0) then b := [op(b),q]; d := [op(d),x]; fi; x := 2*(x-(q*y)); if(member(x,d,'k') and (k > 1) and (b[k] <> b[k-1]) and (q <> floor(x/y))) then s := eval_periodic_confrac_tail(list2runcounts(b[k..nops(b)])); b := b[1..(k-1)]; break; fi; i := i+1; od; if(0 = k) then b := b[1..(nops(b)-1)]; b := [op(b),b[nops(b)]]; fi; if(r < (1/2)) then RETURN(factor(eval_confrac([0,op(list2runcounts(b))],s))); else RETURN(factor(eval_confrac(list2runcounts(b),s))); fi; end;
eval_confrac := proc(c,z) local x,i; x := z; for i in reverse(c) do x := (`if`((0=x),x,(1/x)))+i; od; RETURN(x); end;
eval_periodic_confrac_tail := proc(c) local x,i,u,r; x := (eval_confrac(c,u) - u) = 0; r := [solve(x,u)]; RETURN(max(r[1],r[2])); end;
list2runcounts := proc(b) local a,p,y,c; if(0 = nops(b)) then RETURN([]); fi; a := []; c := 0; p := b[1]; for y in b do if(y <> p) then a := [op(a),c]; c := 0; p := y; fi; c := c+1; od; RETURN([op(a),c]); end;
find_sqrt := proc(x) local n,i,y; n := nops(x); if(n < 2) then RETURN(0); fi; if((2 = n) and (`^` = op(0,x)) and (1/2 = op(2,x))) then RETURN(op(1,x)); else for i from 0 to n do y := find_sqrt(op(i,x)); if(y <> 0) then RETURN(y); fi; od; RETURN(0); fi; end;

Description clarified by Antti Karttunen, Aug 26 2006

A065934 Permutation of N induced by the order-preserving bijection QuQR1toQuQR2 on rationals.

Original entry on oeis.org

1, 5, 13, 2, 23, 25, 3, 9, 20, 11, 95, 49, 6, 223, 57, 4, 39, 80, 10, 45, 92, 47, 383, 97, 12, 415, 208, 55, 3583, 225, 29, 17, 36, 19, 159, 320, 40, 83, 42, 22, 183, 368, 46, 189, 380, 191, 1535, 193, 24, 799, 400, 103, 6655, 3328, 52, 220, 445, 895, 57343, 897

Offset: 1

Antti Karttunen, Dec 07 2001

nonn

This permutation converts the domain between the mappings N2QuQR1 and N2QuQR2 given in A065936 and A065937, i.e. N2QuQR1(j) = N2QuQR2(a[j])

Inverse permutation: A065935. For other needed Maple procedures, see A007305, A047679 and A054424. A065939[n] = a[A065938[n]].

[seq(QuQR1toQuQR2(j),j=1..128)];
QuQR1toQuQR2 := proc(n) local m; m := n + 2^floor_log_2(n); frac2position_in_whole_SB_tree(Q0_1toQ(SternBrocotTreeNum(m)/SternBrocotTreeDen(m))); end;
Q0_1toQ := proc(rr) local r,i; r := rr; i := 0; while(r >= 1/2) do r := 2*(r-(1/2)); i := i+1; od; RETURN(i + (2*r)); end;

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A003285 Period of continued fraction for square root of n (or 0 if n is a square).

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A065936 a(n) is the integer (reduced squarefree) under the square root obtained when the inverse of a variant of Minkowski's question mark function is applied to the n-th ratio A007305(n+1)/A007306(n+1) in the left-hand subtree of Stern-Brocot tree and zero when it results a rational value.

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A065934 Permutation of N induced by the order-preserving bijection QuQR1toQuQR2 on rationals.

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A065939 Position of sqrt(n) in the mapping N2QuQR2 given in A065937.

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