A097845 -id:A097845 - cp's OEIS frontend

A088316 a(n) = 13*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13.

2, 13, 171, 2236, 29239, 382343, 4999698, 65378417, 854919119, 11179326964, 146186169651, 1911599532427, 24996980091202, 326872340718053, 4274337409425891, 55893258663254636, 730886700031736159, 9557420359075824703, 124977351368017457298, 1634262988143302769577

Offset: 0

Nikolay V. Kosinov, Dmitry V. Polyakov (kosinov(AT)unitron.com.ua), Nov 06 2003

For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - Johannes W. Meijer, Jun 12 2010

G. C. Greubel, Table of n, a(n) for n = 0..890
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. A006905, A041318, A054413, A086902, A088316, A097845, A140455.

I:=[2,13]; [n le 2 select I[n] else 13*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022

LinearRecurrence[{13,1}, {2,13}, 31] (* Stefano Spezia, Sep 20 2022 *)

A088316=BinaryRecurrenceSequence(13,1,2,13)
[A088316(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

a(n) = ((13+sqrt(173))/2)^n + ((13-sqrt(173))/2)^n.
Lim_{n -> oo} a(n+1)/a(n) = (13 + sqrt(173))/2.
Lim_{n -> oo} a(n)/a(n+1) = 2/(13+sqrt(173)).
G.f.: (2-13*x)/(1-13*x-x^2). - Philippe Deléham, Nov 02 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2*n+1) = 13*A097845(n).
a(3*n+1) = A041318(5n), a(3n+2) = A041318(5n+3), a(3n+3) = 2*A041318(5n+4).
Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.
Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)

A098244 First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property.

Original entry on oeis.org

1, 170, 29069, 4970629, 849948490, 145336221161, 24851643870041, 4249485765555850, 726637214266180309, 124250714153751276989, 21246145483077202184810, 3632966626892047822325521, 621216047053057100415479281, 106224311079445872123224631530

Offset: 0

Wolfdieter Lang, Sep 10 2004

(13*b(n))^2 - 173*a(n)^2 = -4 with b(n)=A097845(n) give all positive solutions of this Pell equation.

			All positive solutions of Pell equation x^2 - 173*y^2 = -4 are (13=13*1,1), (2236=13*172,170), (382343=13*29411,29069), (65378417=13*5029109,4970629), ...

Indranil Ghosh, Table of n, a(n) for n = 0..446
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 (171,-1).
Index entries for sequences related to Chebyshev polynomials.

a:=[1,170];; for n in [3..20] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,170]; [n le 2 select I[n] else 171*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{171,-1}, {1,170}, 20] (* G. C. Greubel, Aug 01 2019 *)

my(x='x+O('x^20)); Vec((1-x)/(1-171*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-171*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = ((-1)^n)*S(2*n, 13*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-171*x+x^2).
a(n) = S(n, 171) - S(n-1, 171) = T(2*n+1, sqrt(173)/2)/(sqrt(173)/2), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
a(n) = 171*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=170. - Philippe Deléham, Nov 18 2008

A097844 Chebyshev polynomials S(n,171).

Original entry on oeis.org

1, 171, 29240, 4999869, 854948359, 146191169520, 24997835039561, 4274483600595411, 730911697866775720, 124981625851618052709, 21371127108928820237519, 3654337754000976642563040, 624870384807058077058042321, 106849181464252930200282673851

Offset: 0

Wolfdieter Lang, Sep 10 2004

Used for all positive integer solutions of Pell equation x^2 - 173*y^2 = -4. See A097845 with A098244.

Indranil Ghosh, Table of n, a(n) for n = 0..446
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (171,-1).

Cf. A097845, A098244.

a:=[1,171];; for n in [3..30] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 14 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-171*x+x^2) )); // G. C. Greubel, Jan 14 2019

CoefficientList[Series[1/(1-171x+x^2),{x,0,30}],x] (* or *) LinearRecurrence[{171,-1},{1,171},30] (* Harvey P. Dale, Mar 21 2013 *)

my(x='x+O('x^30)); Vec(1/(1-171*x+x^2)) \\ G. C. Greubel, Jan 14 2019

(1/(1-171*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 14 2019

a(n) = S(n, 171) = U(n, 171/2) = S(2*n+1, sqrt(173))/sqrt(173) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x).
a(n) = 171*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=171.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (171+13*sqrt(173))/2 and am = (171-13*sqrt(173))/2 = 1/ap.
G.f.: 1/(1-171*x+x^2).

cp's OEIS Frontend

A088316 a(n) = 13*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13.

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A098244 First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property.

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A097844 Chebyshev polynomials S(n,171).

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