A077232 -id:A077232 - cp's OEIS frontend

A077233 a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= +1 or = -1, with d(n)=A000037(n) (nonsquare). Corresponding smallest b(n)=A077232(n).

1, 1, 1, 2, 3, 1, 1, 3, 2, 5, 4, 1, 1, 4, 39, 2, 12, 42, 5, 1, 1, 5, 24, 13, 2, 273, 3, 4, 6, 1, 1, 6, 4, 3, 5, 2, 531, 30, 24, 3588, 7, 1, 1, 7, 90, 25, 66, 12, 2, 20, 13, 69, 4, 3805, 8, 1, 1, 8, 5967, 4, 936, 30, 413, 2, 125, 5, 3, 6630, 40, 6, 9, 1, 1, 9, 6, 41, 1122, 3, 21, 53, 2, 165, 120, 1260, 221064, 4, 5, 569, 10, 1, 1, 10, 22419

Offset: 1

Wolfdieter Lang, Nov 08 2002

If d(n)=A000037(n) is from A003654 (that is if the regular continued fraction for sqrt(d(n)) has odd (primitive) period length) then the -1 option applies. For such d(n) the minimal b(n) and a(n) numbers for the +1 option are 2*b(n)^2 + 1 and 2*b(n)*a(n), respectively (see Perron I, pp. 94,p5).
For general integer solutions see A077232 comments.
If the trivial solution x=1, y=0 is included, the sequence becomes A006703. - T. D. Noe, May 17 2007

			d=10=A000037(7)=A003654(3), therefore a(7)=1 and b(7)=A077232(7)=3 give 3^2=10*1^2 -1 and 2*b(7)^2+1=19 and 2*b(7)*a(7)=2*3*1=6 satisfy 19^2 - 10*6^2 = +1.
d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) solution of the b^2 - d*a^2 = -1 Pell equation and a(8)=3 and b(8)=A077232(8)=10 satisfy 10^2 - 11*3^2 = +1. See A077232 for further examples.

T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, table p. 301.
O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 26, p. 91 with explanation on pp. 94,95).

Ray Chandler, Table of n, a(n) for n = 1..10000
A. M. Legendre, Fractions les plus simples m/n qui satisfont à l'équation m^2 - an^2 =+-1 pour tout nombre non quarré a depuis 2 jusqu'à 1003, Essai sur la Théorie des Nombres An VI, Table XII. [_Paul Curtz_, Apr 10 2019]

Cf. A000037, A003654, A003814, A033317, A077232.

nmax = 500;
nconv = 200; (* The number of convergents 'nconv' should be increased if the linear recurrence is not found for some terms. *)
nonSquare[n_] := n + Round[Sqrt[n]];
b[n_] := b[n] = Module[{lr}, lr = FindLinearRecurrence[ Numerator[ Convergents[ Sqrt[nonSquare[n]], nconv]]]; (1/2) SelectFirst[lr, #>1&]];
a[n_] := If[n == 1, 1, SelectFirst[{Sqrt[(b[n]^2 - 1)/nonSquare[n]], Sqrt[(b[n]^2 + 1)/nonSquare[n]]}, IntegerQ]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Mar 10 2021 *)

a(n)=sqrt((A077232(n)^2 - (-1)^(c(n)))/A000037(n)) with c(n)=1 if A000037(n)=A003654(k) for some k>=1 else c(n)=0.

A033313 Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y.

Original entry on oeis.org

3, 2, 9, 5, 8, 3, 19, 10, 7, 649, 15, 4, 33, 17, 170, 9, 55, 197, 24, 5, 51, 26, 127, 9801, 11, 1520, 17, 23, 35, 6, 73, 37, 25, 19, 2049, 13, 3482, 199, 161, 24335, 48, 7, 99, 50, 649, 66249, 485, 89, 15, 151, 19603, 530, 31, 1766319049, 63, 8, 129, 65, 48842, 33

Offset: 1

Eric W. Weisstein

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris. See column C page 19.
H. W. Lenstra, jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.
F. Richman and R. Mines, Pell's equation
Derek Smith, Historical Overview of Pell Equations
Derek Smith, The Search For An Exhaustive Solution to Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation

See A033317 (for y's).

Cf. A000037, A002350, A077232, A077233.

F:= proc(d) local r,Q; uses numtheory;
  Q:= cfrac(sqrt(d),'periodic','quotients'):
  r:= nops(Q[2]);
  if r::odd then
    numer(cfrac([op(Q[1]),op(Q[2]),op(Q[2][1..-2])]))
  else
    numer(cfrac([op(Q[1]),op(Q[2][1..-2])]));
  fi
end proc:
map(F, remove(issqr,[$1..100])); # Robert Israel, May 17 2015

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
A033313 = DeleteCases[PellSolve /@ Range[100], {}][[All, 1]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002350 *)
Table[If[! IntegerQ[Sqrt[k]], {k,FindInstance[x^2 - k*y^2 == 1 && x > 0 && y > 0, {x, y},Integers]}, Nothing], {k, 2, 80}][[All, 2, 1, 1, 2]] (* Horst H. Manninger, Mar 28 2021 *)

a(n) = sqrt(1 + A000037(n)*A033317(n)^2), or

a(n) = sqrt(1 + (n + floor(1/2 + sqrt(n)))*A033317(n)^2). - Zak Seidov, Oct 24 2013

Offset switched to 1 by R. J. Mathar, Sep 21 2009

Name corrected by Wolfdieter Lang, Sep 03 2015

A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

Original entry on oeis.org

2, 1, 4, 2, 3, 1, 6, 3, 2, 180, 4, 1, 8, 4, 39, 2, 12, 42, 5, 1, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3, 6630, 40, 6, 9

Offset: 1

Eric W. Weisstein

nonn

D = D(n) = A000037(n). - Wolfdieter Lang, Oct 04 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Laurent Beeckmans, Squares Expressible as Sum of Consecutive Squares, Am. Math. Monthly, Volume 101, Number 5, page 442, May 1994.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris. See column B page 19.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Pell Equation

Cf. A000037, A033313 (for the x's), A077232, A077233.

F:= proc(d) local r,Q; uses numtheory;
  Q:= cfrac(sqrt(d),'periodic','quotients'):
  r:= nops(Q[2]);
  if r::odd then
    denom(cfrac([op(Q[1]),op(Q[2]),op(Q[2][1..-2])]))
  else
    denom(cfrac([op(Q[1]),op(Q[2][1..-2])]));
  fi
end proc:
map(F, remove(issqr,[$1..100])); # Robert Israel, May 17 2015

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
A033317 = DeleteCases[PellSolve /@ Range[100], {}][[All, 2]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002349 *)

a(n) = sqrt((A033313(n)^2 - 1)/A000037(n)). - Jinyuan Wang, Jul 09 2020

A006702 Solution to a Pellian equation: least x such that x^2 - n*y^2 = +- 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 8, 3, 1, 3, 10, 7, 18, 15, 4, 1, 4, 17, 170, 9, 55, 197, 24, 5, 1, 5, 26, 127, 70, 11, 1520, 17, 23, 35, 6, 1, 6, 37, 25, 19, 32, 13, 3482, 199, 161, 24335, 48, 7, 1, 7, 50, 649, 182, 485, 89, 15, 151, 99, 530, 31, 29718, 63, 8, 1, 8, 65, 48842

Offset: 1

N. J. A. Sloane

When n is a square, the trivial solution x=1, y=0 is taken; otherwise we take the least x that satisfies either the +1 or -1 equation. - T. D. Noe, May 19 2007
Apparently the generating function of the sequence of 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). If so, then it seems that a(n) is half the value of C for each nonsquare n, or 1. See A003285 for the conjecture regarding m. The same conjectures apply to the sequences of the numerators of continued fraction convergents to sqrt(n). - Ralf Stephan, Dec 12 2013
The conjecture is true, cf. link. - Jan Ritsema van Eck, Mar 08 2021

A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443. (Annotated scanned copy)
Jan Ritsema van Eck, Proof of conjecture in A006702, Mar 08 2021
M. Zuker, Fundamental solution to Pell's Equation x^2 - d*y^2 = +-1

Cf. A006703, A077232.

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 *)

Corrected and extended by T. D. Noe, May 19 2007

A078988 Chebyshev sequence with Diophantine property.

Original entry on oeis.org

1, 65, 4289, 283009, 18674305, 1232221121, 81307919681, 5365090477825, 354014663616769, 23359602708228929, 1541379764079492545, 101707704826538279041, 6711167138787446924161, 442835323455144958715585, 29220420180900779828304449, 1928104896615996323709378049

Offset: 0

Wolfdieter Lang, Jan 10 2003

Bisection (even part) of A041025.
(4*A078989(n))^2 - 17*a(n)^2 = -1 (Pell -1 equation, see A077232-3).
Starting with a(1), hypotenuses of primitive Pythagorean triples in A195619 and A195620. - Clark Kimberling, Sep 22 2011

			(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.

Colin Barker, Table of n, a(n) for n = 0..549
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (66,-1).

Row 66 of array A094954.
Cf. A097316 for S(n, 66).
Row 4 of array A188647.

a:=[1,65];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1, 65]; [n le 2 select I[n] else 66*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

CoefficientList[Series[(1-x)/(1-66x+x^2), {x,0,20}], x] (* Michael De Vlieger, Apr 15 2019 *)
LinearRecurrence[{66,-1}, {1,65}, 21] (* G. C. Greubel, Aug 01 2019 *)

Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ Colin Barker, Jun 15 2015

((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

G.f.: (1-x)/(1-66*x+x^2).
a(n) = T(2*n+1, sqrt(17))/sqrt(17) = ((-1)^n)*S(2*n, 8*i) = S(n, 66) - S(n-1, 66) with i^2=-1 and T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
a(n) = A041025(2*n).
a(n) = 66*a(n-1) - a(n-2) for n>1 ; a(0)=1, a(1)=65. - Philippe Deléham, Nov 18 2008

A078989 Chebyshev sequence with Diophantine property.

Original entry on oeis.org

1, 67, 4421, 291719, 19249033, 1270144459, 83810285261, 5530208682767, 364909962777361, 24078527334623059, 1588817894122344533, 104837902484740116119, 6917712746098725319321, 456464203340031130959067, 30119719707695955917979101, 1987445036504593059455661599

Offset: 0

Wolfdieter Lang, Jan 10 2003

One fourth of bisection (even part) of A041024.

(4*a(n))^2 - 17*A078988(n)^2= -1 (Pell -1 equation, see A077232-3).

			(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..548
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for linear recurrences with constant coefficients, signature (66, -1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A097316 for S(n, 66).
Cf. A041024.
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

a:=[1,67];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Apr 05 2018

LinearRecurrence[{66, -1}, {1, 67}, 20] (* Bruno Berselli, Apr 03 2018 *)

x='x+O('x^99); Vec((1+x)/(1-66*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 66*x + x^2).
a(n) = 66*a(n-1) - a(n-2) for n>=1, a(-1)=-1, a(0)=1.
a(n) = S(2*n, 2*sqrt(17)) = -i*((-1)^n)*T(2*n+1, 4*i)/4 = S(n, 66) + S(n-1, 66) with i^2=-1 and S(n, x), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = A041024(2*n)/4.
a(n) = (1/4)*sinh((2*n + 1)*arcsinh(4)). - Bruno Berselli, Apr 03 2018

A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.

Original entry on oeis.org

0, 5, 120, 2760, 63365, 1454640, 33393360, 766592645, 17598237480, 403992869400, 9274237758725, 212903475581280, 4887505700610720, 112199727638465285, 2575706229984090840, 59129043561995624040, 1357392295695915262085, 31160893757444055403920

Offset: 1

Paul Weisenhorn, May 24 2009

The ansatz 3*a(n)+1=A^2, 7*a(n)+1=B^2 is equivalent to the Pell equation x^2-21*y^2=1 (see A077232 for d=21), with x=(21*a(n)+5)/2 and y=A*B/2.
The associated A are in A004253, the B in A030221.
Bisection of A089927. - R. J. Mathar, Jul 10 2009

Michael De Vlieger, Table of n, a(n) for n = 1..736
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A089927, A160682, A245031, A334673.

j:=0: for n from 0 to 1000000 do a:=sqrt(3*n+1): b:=sqrt(7*n+1):
if (trunc(a)=a) and (trunc(b)=b) then j:=j+1: print(j,n,a,b): end if:
end do:

LinearRecurrence[{24,-24,1},{0,5,120},30] (* Harvey P. Dale, Dec 17 2013 *)

a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3).
a(n) = (A004253(n)^2 - 1)/3 = (A030221(n)^2 - 1)/7.
a(n) = ((5+w)/2*((23+5*w)/2)^(n-1) + (5-w)/2*((23-5*w)/2)^(n-1) - 5)/21; where w=sqrt(21). [Corrected by Kevin Ryde, Sep 11 2020]
G.f.: 5*x^2/((1-x)*(x^2-23*x+1)). - R. J. Mathar, Jul 10 2009
From Francesca Arici, Sep 12 2020: (Start)
a(n) = 23*a(n-1) - a(n-2) + 5.
a(n) = A004254(n)* A004254(n+1). (End)
a(n) = 5*A334673(n-1). - Hugo Pfoertner, Apr 07 2021

Edited and extended by R. J. Mathar, Jul 10 2009

Name edited by Michel Marcus, Sep 12 2020

cp's OEIS Frontend

A077233 a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= +1 or = -1, with d(n)=A000037(n) (nonsquare). Corresponding smallest b(n)=A077232(n).

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A033313 Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y.

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A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

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A006702 Solution to a Pellian equation: least x such that x^2 - n*y^2 = +- 1.

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A078988 Chebyshev sequence with Diophantine property.

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A078989 Chebyshev sequence with Diophantine property.

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GAP

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A160695 Integers m such that 3*m+1 and 7*m+1 are both perfect squares.

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A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.