A188647 -id:A188647 - cp's OEIS frontend

A097736 Pell equation solutions (8b(n))^2 - 65a(n)^2 = -1 with b(n):=A097735(n), n >= 0.

1, 257, 66305, 17106433, 4413393409, 1138638393089, 293764292023553, 75790048703683585, 19553538801258341377, 5044737220675948391681, 1301522649395593426712321, 335787798806842428143387137, 86631950569515950867567169025, 22350707459136308481404186221313

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..413
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 (258,-1).

Cf. A097734 for S(n, 258).

Row 8 of array A188647.

a:=[1,257];; for n in [3..20] do a[n]:=258*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,257]; [n le 2 select I[n] else 258*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{258, -1},{1, 257},20] (* Ray Chandler, Aug 12 2015 *)

my(x='x+O('x^20)); Vec((1-x)/(1-258*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-258*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 2*129) - S(n-1, 2*129) = T(2*n+1, sqrt(65))/sqrt(65), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 16*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-258*x+x^2).
a(n) = 258*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=257. - Philippe Deléham, Nov 18 2008

A097739 Pell equation solutions (9b(n))^2 - 82a(n)^2 = -1 with b(n):=A097738(n), n >= 0.

Original entry on oeis.org

1, 325, 105949, 34539049, 11259624025, 3670602893101, 1196605283526901, 390089651826876625, 127168029890278252849, 41456387654578883552149, 13514655207362825759747725, 4405736141212626618794206201, 1436256467380108914901151473801, 468215202629774293631156586252925

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (9*1=9;1), (2943=9*327;325), (959409=9*106601;105949), ... give the positive integer solutions to x^2 - 82*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..397
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 (326,-1).

Cf. A097737 for S(n, 326).

Row 9 of array A188647.

a:=[1,325];; for n in [3..20] do a[n]:=326*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,325]; [n le 2 select I[n] else 326*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{326, -1},{1, 325},12] (* Ray Chandler, Aug 12 2015 *)

my(x='x+O('x^20)); Vec((1-x)/(1-326*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-326*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 2*163) - S(n-1, 2*163) = T(2*n+1, sqrt(82))/sqrt(82), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 18*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1- 326*x+x^2).
a(n) = 326*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=325. - Philippe Deléham, Nov 18 2008

A097742 Pell equation solutions (10b(n))^2 - 101a(n)^2 = -1 with b(n)=A097741(n), n >= 0.

Original entry on oeis.org

1, 401, 161201, 64802401, 26050404001, 10472197606001, 4209797387208401, 1692328077460171201, 680311677341601614401, 273483601963246388818001, 109939727677547706703222001, 44195497042772214848306426401, 17766479871466752821312480191201, 7142080712832591861952768730436401

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (10*1=10;1), (4030=10*403;401), (1620050=10*162005;161201), ... give the positive integer solutions to x^2 - 101*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..383
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 (402,-1).

Cf. A097740 for S(n, 402).

Row 10 of array A188647.

a:=[1,401];; for n in [3..20] do a[n]:=402*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,401]; [n le 2 select I[n] else 402*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{402, -1},{1, 401},12] (* Ray Chandler, Aug 12 2015 *)

my(x='x+O('x^20)); Vec((1-x)/(1-402*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-402*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 2*201) - S(n-1, 2*201) = T(2*n+1, sqrt(101))/sqrt(101), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 20*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-402*x+x^2).
a(n) = 402*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=401. - Philippe Deléham, Nov 18 2008

A097767 Pell equation solutions (11b(n))^2 - 122a(n)^2 = -1 with b(n):=A097766(n), n >= 0.

Original entry on oeis.org

1, 485, 235709, 114554089, 55673051545, 27056988496781, 13149640736384021, 6390698340894137425, 3105866244033814404529, 1509444603902092906463669, 733586971630173118726938605, 356521758767660233608385698361

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (11*1=11;1), (5357=11*487;485), (2603491=11*236681;235709), ... give the positive integer solutions to x^2 - 122*y^2 =-1.

Seiichi Manyama, Table of n, a(n) for n = 0..372
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 (486,-1).

Cf. A097765 for S(n, 486).

Row 11 of array A188647.

a:=[1,485];; for n in [3..20] do a[n]:=486*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,485]; [n le 2 select I[n] else 486*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{486, -1},{1, 485},20] (* Ray Chandler, Aug 12 2015 *)

my(x='x+O('x^20)); Vec((1-x)/(1-486*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-486*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 2*243) - S(n-1, 2*243) = T(2*n+1, sqrt(122))/sqrt(122), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 22*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-486*x+x^2).
a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=485. - Philippe Deléham, Nov 18 2008

A097770 Pell equation solutions (12b(n))^2 - 145a(n)^2 = -1 with b(n)=A097769(n), n >= 0.

Original entry on oeis.org

1, 577, 333505, 192765313, 111418017409, 64399421297089, 37222754091700033, 21514687465581321985, 12435452132351912407297, 7187669817811939790095681, 4154460719243168846762896321, 2401271108052733781489163977857

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (12*1=12;1), (6948=12*579;577), (4015932=12*334661;333505), ... give the positive integer solutions to x^2 - 145*y^2 =-1.

Seiichi Manyama, Table of n, a(n) for n = 0..362
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 (578,-1).

Cf. A097768 for S(n, 486).

Row 12 of array A188647.

a:=[1,577];; for n in [3..20] do a[n]:=578*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,577]; [n le 2 select I[n] else 578*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{578, -1},{1, 577},12] (* Ray Chandler, Aug 12 2015 *)

my(x='x+O('x^20)); Vec((1-x)/(1-578*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-578*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 2*289) - S(n-1, 2*289) = T(2*n+1, sqrt(145))/sqrt(145), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 24*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-578*x+x^2).
a(n) = 578*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=577. - Philippe Deléham, Nov 18 2008

A097773 Pell equation solutions (13b(n))^2 - 170a(n)^2 = -1 with b(n):=A097772(n), n >= 0.

Original entry on oeis.org

1, 677, 459005, 311204713, 210996336409, 143055204880589, 96991217912702933, 65759902689607707985, 44585117032336113310897, 30228643588021195217080181, 20494975767561338021067051821, 13895563341762999157088244054457

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (13*1=13;1), (8827=13*679;677), (5984693=13*460361;459005), ... give the positive integer solutions to x^2 - 170*y^2 =-1.

Seiichi Manyama, Table of n, a(n) for n = 0..353
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 (678,-1).

Cf. A097771 for S(n, 678).

Row 13 of array A188647.

a:=[1,677];; for n in [3..20] do a[n]:=678*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,677]; [n le 2 select I[n] else 678*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{678, -1},{1, 677},11] (* Ray Chandler, Aug 12 2015 *)

my(x='x+O('x^20)); Vec((1-x)/(1-678*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-678*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = ((-1)^n)*S(2*n, 26*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-678*x+x^2).
a(n) = S(n, 2*339) - S(n-1, 2*339) = T(2*n+1, sqrt(170))/sqrt(170), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = 678*a(n-1) - a(n-2), n>1; a(0)=1, a(1)=677. - Philippe Deléham, Nov 18 2008

A323012 a(n) = (1/sqrt(n^2+1)) * T_{2*n+1}(sqrt(n^2+1)) where T_{n}(x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 5, 305, 53353, 18674305, 10928351501, 9616792908241, 11868363584907985, 19553538801258341377, 41456387654578883552149, 109939727677547706703222001, 356521758767660233608385698361, 1387930545993760882531890016305025

Offset: 0

Seiichi Manyama, Jan 02 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..193
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Diagonal of A188647.

[&+[Binomial(2*n+1,2*k)*(n^2+1)^(n-k)*n^(2*k): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Jan 03 2019

Join[{1}, Table[Sum[Binomial[2 n + 1, 2 k] (n^2 + 1)^(n - k) n^(2 k), {k, 0, n}], {n, 20}]] (* Vincenzo Librandi, Jan 03 2019 *)

{a(n) = sum(k=0, n, binomial(2*n+1, 2*k)*(n^2+1)^(n-k)*n^(2*k))}

a(n) = Sum_{k=0..n} binomial(2*n+1,2*k)*(n^2+1)^(n-k)*n^(2*k).

a(n) ~ 2^(2*n) * n^(2*n). - Vaclav Kotesovec, Jan 02 2019

cp's OEIS Frontend

A097736 Pell equation solutions (8*b(n))^2 - 65*a(n)^2 = -1 with b(n):=A097735(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A097739 Pell equation solutions (9*b(n))^2 - 82*a(n)^2 = -1 with b(n):=A097738(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A097742 Pell equation solutions (10*b(n))^2 - 101*a(n)^2 = -1 with b(n)=A097741(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A097767 Pell equation solutions (11*b(n))^2 - 122*a(n)^2 = -1 with b(n):=A097766(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A097770 Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n)=A097769(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A097773 Pell equation solutions (13*b(n))^2 - 170*a(n)^2 = -1 with b(n):=A097772(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

A097736 Pell equation solutions (8b(n))^2 - 65a(n)^2 = -1 with b(n):=A097735(n), n >= 0.

A097739 Pell equation solutions (9b(n))^2 - 82a(n)^2 = -1 with b(n):=A097738(n), n >= 0.

A097742 Pell equation solutions (10b(n))^2 - 101a(n)^2 = -1 with b(n)=A097741(n), n >= 0.

A097767 Pell equation solutions (11b(n))^2 - 122a(n)^2 = -1 with b(n):=A097766(n), n >= 0.

A097770 Pell equation solutions (12b(n))^2 - 145a(n)^2 = -1 with b(n)=A097769(n), n >= 0.

A097773 Pell equation solutions (13b(n))^2 - 170a(n)^2 = -1 with b(n):=A097772(n), n >= 0.