A077428 -id:A077428 - cp's OEIS frontend

A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)b(n) - G(n)b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

1, 2, 5, 3, 3, 27, 7, 37, 4, 4, 171, 22, 9, 14, 1193, 5, 5, 553, 16, 6173, 11, 45, 143, 849, 6, 6, 18339, 94, 1893, 103, 13, 33, 2353, 115, 12703, 7, 7, 67115, 701, 73, 59, 1891117, 15, 551427, 23, 49771, 39, 4105015, 8, 8, 24673, 41, 75585293, 25, 9095891, 989, 17, 386, 6445, 87, 771, 1385

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

This equation can also be written as (2*a(n) - b(n))^2 - D(n)*b(n)^2 = +4 or -4 with D(n) := A077425(n) = 1 + 4*G(n).
This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations.
For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Wolfdieter Lang, Pairs of smallest positive solutions for +1 and -1 if they exist.

Cf. A077427, A217470.

g[n_] := Ceiling[Sqrt[n]] + n - 1; r[n_] := Reduce[an > 0 && bn > 0 && (an ^2 - an*bn - g[n]*bn^2 == 1 || an^2 - an*bn - g[n]*bn^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}], 1], an | bn]; a[n_] := a[n] = Min[ab[n][[All, 1]]]; Table[Print[{n, a[n]}]; a[n], {n, 1, 62}] (* Jean-François Alcover, Oct 04 2012 *)

a(n) = (A078361(n) + A077058(n)) / 2. [Max Alekseyev, Feb 06 2010]

More terms from Max Alekseyev, Feb 06 2010

a(9), a(33), a(54) corrected (after notice by Jean-François Alcover); a(58) through a(62) added. - Wolfdieter Lang, Oct 04 2012

A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)a(n) - G(n)a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 8, 2, 10, 1, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2, 8, 25, 146, 1, 1, 2968, 15, 298, 16, 2, 5, 352, 17, 1856, 1, 1, 9384, 97, 10, 8, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 3, 1084152, 117, 2, 45, 746, 10, 88, 157, 126890, 1, 1

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

This equation can also be written as (2*b(n)-a(n))^2 - D(n)*a(n)^2 = +4 or -4 with D(n) := A077425(n)=1+4*G(n).
This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations.
For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

g[n_] := Ceiling[ Sqrt[n] ] + n - 1; r[n_] := Reduce[an > 0 && (bn^2 - bn *an - g[n]*an^2 == 1 || bn^2 - bn *an - g[n]*an^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}] , 1] , an | bn]; a[n_] := a[n] = Min[ ab[n][[All, 1]] ]; Table[ Print[{n, a[n]}]; a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 03 2012 *)

forstep(D=1,1000,4, if(issquare(D),next); u=bnfinit(x^2-D).fu[1]; k=1; while( denominator(t=polcoeff(lift(u^k),1)*2)>1, k++); print1(abs(t),", "); ) \\ Max Alekseyev, Feb 06 2010

More terms from Max Alekseyev, Feb 06 2010

A078363 A Chebyshev T-sequence with Diophantine property.

Original entry on oeis.org

2, 13, 167, 2158, 27887, 360373, 4656962, 60180133, 777684767, 10049721838, 129868699127, 1678243366813, 21687295069442, 280256592535933, 3621648407897687, 46801172710133998, 604793596823844287, 7815515585999841733, 100996909021174098242

Offset: 0

Wolfdieter Lang, Nov 29 2002

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 = +4 with companion sequence b(n)=A078362(n-1), n>=1.

Except for the first term, positive values of x (or y) satisfying x^2 - 13xy + y^2 + 165 = 0. - Colin Barker, Feb 26 2014

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (13,-1).

Cf. A078362.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A049310, A053120.

a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{13,-1},{2,13},20] (* Harvey P. Dale, Oct 28 2016 *)

a(n)=if(n<0,0,2*subst(poltchebi(n),x,13/2))

a(n)=if(n<0,0,polsym(1-13*x+x^2,n)[n+1])

Vec((2-13*x)/(1-13*x+x^2) + O(x^100)) \\ Colin Barker, Feb 26 2014

[lucas_number2(n,13,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

a(n) = 13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2.
a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-13*x)/(1-13*x+x^2).
a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.
a(n) = sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).
E.g.f.: 2*exp(13*x/2)*cosh(sqrt(165)*x/2). - Stefano Spezia, Sep 24 2022

More terms from Colin Barker, Feb 26 2014

A078368 A Chebyshev S-sequence with Diophantine property.

Original entry on oeis.org

1, 19, 360, 6821, 129239, 2448720, 46396441, 879083659, 16656193080, 315588584861, 5979526919279, 113295422881440, 2146633507828081, 40672741225852099, 770635449783361800, 14601400804658022101

Offset: 0

Wolfdieter Lang, Nov 29 2002

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 357*a^2 =+4 with companion sequence b(n)=A078369(n+1), n>=0.
This is the m=21 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..20 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364, A077412, A078366 and A049660. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 19's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,18}. Milan Janjic, Jan 25 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=19, q=-1.
Tanya Khovanova, Recursive Sequences
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=21.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (19,-1).

a(n) = sqrt((A078369(n+1)^2 - 4)/357), n>=0, (Pell equation d=357, +4).

Cf. A077428, A078355 (Pell +4 equations).

Join[{a=1,b=19},Table[c=19*b-a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{19,-1},{1,19},20] (* Harvey P. Dale, Feb 10 2019 *)

[lucas_number1(n,19,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

a(n) = 19*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n) = (ap^(n+1)-am^(n+1))/(ap-am) with ap = (19+sqrt(357))/2 and am = (19-sqrt(357))/2.
a(n) = S(2*n+1, sqrt(21))/sqrt(21) = S(n, 19); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
G.f.: 1/(1-19*x+x^2).
a(n) = Sum_{k=0..n} A101950(n,k)*18^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/17*(17 + sqrt(357)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/38*(17 + sqrt(357)). - Peter Bala, Dec 23 2012

A090733 a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.

Original entry on oeis.org

2, 25, 623, 15550, 388127, 9687625, 241802498, 6035374825, 150642568127, 3760028828350, 93850078140623, 2342491924687225, 58468448039040002, 1459368709051312825, 36425749278243780623, 909184363247043202750

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 69*(3*b)^2 =+4 together with the companion sequence b(n)=A097780(n-1), n>=0.

			(x,y) =(2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Indranil Ghosh, Table of n, a(n) for n = 0..714
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (25,-1)

Cf. A046069, A082974.
a(n)=sqrt(4 + 69*(3*A097780(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A097779 for 2*T(n, 23/2).

a[0] = 2; a[1] = 25; a[n_] := 25a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

[lucas_number2(n,25,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = S(n, 25) - S(n-2, 25) = 2*T(n, 25/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 25)=A097780(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.
G.f.: (2-25*x)/(1-25*x+x^2).

Extension, Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

A078369 A Chebyshev T-sequence with Diophantine property.

Original entry on oeis.org

2, 19, 359, 6802, 128879, 2441899, 46267202, 876634939, 16609796639, 314709501202, 5962870726199, 112979834296579, 2140653980908802, 40559445802970659, 768488816275533719, 14560728063432170002

Offset: 0

Wolfdieter Lang, Nov 29 2002

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 357*b^2 =+4 with companion sequence b(n)=A078368(n-1), n>=1.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (19,-1).

a(n)=sqrt(4 + 357*A078368(n-1)^2), n>=1, (Pell equation d=357, +4).

Cf. A077428, A078355 (Pell +4 equations).

a[0] = 2; a[1] = 19; a[n_] := 19a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{19,-1},{2,19},20] (* Harvey P. Dale, Dec 24 2021 *)

[lucas_number2(n,19,1) for n in range(0,20)] # Zerinvary Lajos, Jun 27 2008

a(n)=19*a(n-1)-a(n-2), n >= 1; a(-1)=19, a(0)=2.
a(n) = S(n, 19) - S(n-2, 19) = 2*T(n, 19/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 19)=A078368(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-19*x)/(1-19*x+x^2).
a(n) = ap^n + am^n, with ap := (19+sqrt(357))/2 and am := (19-sqrt(357))/2.

A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.

Original entry on oeis.org

2, 21, 439, 9198, 192719, 4037901, 84603202, 1772629341, 37140612959, 778180242798, 16304644485799, 341619353958981, 7157701788652802, 149970118207749861, 3142214780574094279, 65836540273848229998

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 437*b^2 =+4 with companion sequence b(n)=A092499(n), n>=0.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Indranil Ghosh, Table of n, a(n) for n = 0..755
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (21,-1).

Cf. A085985.
a(n)=sqrt(4 + 437*A092499(n)^2), n>=1, (Pell equation d=437, +4).
Cf. A077428, A078355 (Pell +4 equations).

a[0] = 2; a[1] = 21; a[n_] := 21a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

[lucas_number2(n,21,1) for n in range(0,20)] # Zerinvary Lajos, Jun 27 2008

a(n) = S(n, 21) - S(n-2, 21) = 2*T(n, 21/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 21)=A092499(n+1). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
G.f.: (2-21*x)/(1-21*x+x^2).

Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004

A090251 a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29.

Original entry on oeis.org

2, 29, 839, 24302, 703919, 20389349, 590587202, 17106639509, 495501958559, 14352450158702, 415725552643799, 12041688576511469, 348793243166188802, 10102962363242963789, 292637115290879761079, 8476373381072270107502

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004

a(n+1)/a(n) converges to ((29+sqrt(837))/2) =28.9654761... Lim a(n)/a(n+1) as n approaches infinity = 0.0345238... =2/(29+sqrt(837)) =(29-sqrt(837))/2. Lim a(n+1)/a(n) as n approaches infinity = 28.9654761... = (29+sqrt(837))/2=2/(29-sqrt(837)). Lim a(n)/a(n+1) = 29 - Lim a(n+1)/a(n).
A Chebyshev T-sequence with a Diophantine property.
a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 93*(3*b)^2 =+4 with companion sequence b(n)=A097782(n+1), n>=0.

			a(4) =703919 = 29a(3) - a(2) = 29*24302 - 839= ((29+sqrt(837))/2)^4 + ((29-sqrt(837))/2)^4 = 703918.99999857 + 0.00000142 =703919.
(x,y) = (2;0), (29;1), (839;29), (24302,840), ..., give the
nonnegative integer solutions to x^2 - 93*(3*y)^2 =+4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (29, -1).

Cf. A083148, A007610.
a(n)=sqrt(4 + 93*(3*A097782(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A090248 for 2*T(n, 27/2).

a[0] = 2; a[1] = 29; a[n_] := 29a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{29,-1},{2,29},30] (* Harvey P. Dale, May 28 2013 *)

[lucas_number2(n,29,1) for n in range(0,16)] # Zerinvary Lajos, Jun 27 2008

a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29. a(n) = ((29+sqrt(837))/2)^n + ((29-sqrt(837))/2)^n, (a(n))^2 =a(2n)+2.
a(n) = S(n, 29) - S(n-2, 29) = 2*T(n, 29/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 27)=A097781(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (29+3*sqrt(93))/2 and am := (29-3*sqrt(93))/2.
G.f.: (2-29*x)/(1-29*x+x^2).

More terms from Robert G. Wilson v, Jan 30 2004

Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

A078361 Minimal positive solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 or -4 with D(n)=A077425(n). The companion sequence is b(n)=A077058(n).

Original entry on oeis.org

1, 3, 8, 5, 5, 46, 12, 64, 7, 7, 302, 39, 16, 25, 2136, 9, 9, 1000, 29, 11208, 20, 82, 261, 1552, 11, 11, 33710, 173, 3488, 190, 24, 61, 4354, 213, 23550, 13, 13, 124846, 1305, 136, 110, 3528264, 28, 1030190, 43, 93102, 73, 7688126, 15, 15, 46312, 77

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

Computed from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values for the Diophantine eq. x^2 - x*y - ((D(n)-1)/4)*y^2= +1, resp., -1 if D(n)=A077425(n), resp, D(n)=A077425(n) and D(n) also in A077426 (this second case excludes in Perron's table the D values with a 'Teilnenner' in brackets).
The conversion from the x,y values of Perron's table to the minimal a=a(n) and b=b(n) solutions is a(n)=2*x(n)-y(n) and b(n)=y(n). If D(n)=A077425(n) is not in A077426 then the equation with -4 has no solution and a(n) and b(n) are the minimal solutions of the a(n)^2 - D(n)*b(n)^2 = +4 equation. If D(n)=A077425(n) is in A077426 then the a(n) and b(n) values are the minimal solution of the a(n)^2 - D(n)*b(n)^2 = -4 equation. In this case a(+,n)= a(n)^2+2 and b(+,n)=a(n)*b(n) are the minimal solution of a^2 - D(n)*b^2 = +4.
For Pell equation a^2 - D*b^2 = +4, see A077428 and A078355. For Pell equation a^2 - D*b^2 = -4, see A078356 and A078357.

			29=D(5)=A077425(5) is A077426(4), hence a(5)=5 and b(5)=A077058(5)=1 solve a^2 - 29*b^2=-4 minimally and a(+,5)=a(5)^2+2=27 with b(+,5)=a(5)*b(5)=5*1=5 solve a^2 - 29*b^2=+4 minimally. See also A077428 with companion A078355.
21=D(4)=A077425(4) is not in A077426, hence a(4)=5 and b(4)=A077058(4)=1 give the solution with minimal positive b of a^2 - 21*b^2=+4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

More terms from Matthew Conroy, Apr 20 2003

A240469 Values k where the maximum number of distinct rational solutions to x^2 - Dy^2 = t, 0 < D <= k, 0 < t <= k, achieves a new record.

Original entry on oeis.org

1, 2, 7, 10, 17, 32, 73, 144, 241, 336, 360, 720, 1080, 1260

Offset: 1

Ralf Stephan, Apr 06 2014

Record values are in A240470.

			All Diophantine equations x^2 - Dy^2 = t, 0 < D <= 16, 0 < t <= 16, D squarefree, have fewer than 4 distinct solutions; the first with 4 solutions is x^2 - 17y^2 = 16 with the solutions (x,y) = (9/2,1/2), (21,5), (4,0), (13,3), so 17 is in the sequence.

Cf. A218459, A077428, A035251, A031363.

{ r(l,k)=if(!issquarefree(l)||!polisirreducible(z^2-l),return(0));v=bnfisintnorm(bnfinit(z^2-l), k);if(!#v,return(0));s=0;for(k=1,#v,p=v[k];a=polcoeff(p,0);b=polcoeff(p,1);f=1;for(l=k+1,#v,p=v[l];aa=polcoeff(p,0);bb=polcoeff(p,1);if(abs(a)==abs(aa)&&abs(b)==abs(bb),f=0;break));s=s+f);s
m=0;n=0;while(1,n=n+1;res=0;for(l=1,n,rr=r(l,n);if(rr>res,res=rr));for(k=1,n-1,rr=r(n,k);if(rr>res,res=rr));if(res>m,m=res;print(n,","))) }

cp's OEIS Frontend

A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)*b(n) - G(n)*b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

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A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)*a(n) - G(n)*a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).

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A078363 A Chebyshev T-sequence with Diophantine property.

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A078368 A Chebyshev S-sequence with Diophantine property.

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A090733 a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.

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A078369 A Chebyshev T-sequence with Diophantine property.

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A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.

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A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)b(n) - G(n)b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)a(n) - G(n)a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n).