A088317 -id:A088317 - cp's OEIS frontend

A086594 a(n) = 8*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=8.

2, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, 153992264, 1250895426, 10161155672, 82540140802, 670482282088, 5446398397506, 44241669462136, 359379754094594, 2919279702218888, 23713617371845698, 192628218676984472, 1564739366787721474

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 11 2003

a(n+1)/a(n) converges to 4 + sqrt(17).

			a(4) = 8*a(3)+a(2) = 8*536+66 = 4354.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (8,1).

Cf. A003285.

I:=[2,8]; [n le 2 select I[n] else 8*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

LinearRecurrence[{8,1},{2,8},30] (* Harvey P. Dale, Sep 21 2014 *)
RecurrenceTable[{a[0] == 2, a[1] == 8, a[n] == 8 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)

x='x+O('x^30); Vec(2*(1-4*x)/(1-8*x-x^2)) \\ G. C. Greubel, Nov 07 2018

a(n) = (4+sqrt(17))^n + (4-sqrt(17))^n.
O.g.f: 2*(-1+4*x)/(-1+8*x+x^2). - R. J. Mathar, Dec 02 2007
a(n) = 2*A088317(n). - R. J. Mathar, Sep 27 2014

A099370 Chebyshev polynomial of the first kind, T(n,x), evaluated at x=33.

Original entry on oeis.org

1, 33, 2177, 143649, 9478657, 625447713, 41270070401, 2723199198753, 179689877047297, 11856808685922849, 782369683393860737, 51624542295308885793, 3406437421806992601601, 224773245296966202819873

Offset: 0

Wolfdieter Lang, Oct 18 2004

Used in A099369.
Solutions of the Pell equation x^2 - 17y^2 = 1 (x values). After initial term this sequence bisects A041024. See 8*A097316(n-1) with A097316(-1) = 0 for corresponding y values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2. (See related comments in A088317, which this sequence also bisects.). - Rick L. Shepherd, Jul 31 2006
From a(n) = T(n, 33) (see the formula section) and the de Moivre-Binet formula for T(n,x=33) follows a(n+1)/a(n) = 33  + 8*sqrt(17), which is the conjectured value (4+sqrt(17))^2 given in the previous comment by Rick L. Shepherd. - Wolfdieter Lang, Jun 28 2013
Also numbers k such that 17*(k-1)*(k+1) is a square. - Bruno Berselli, May 31 2025

			a(1)^2 - 17*A121470(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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
Eric Weisstein's World of Mathematics, Pell Equation
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (66,-1).

Cf. A121470, A041024, A040012.

Row 4 of array A188645.

LinearRecurrence[{66, -1},{1, 33},14] (* Ray Chandler, Aug 11 2015 *)

\\ Program uses fact that continued fraction for sqrt(17) = [4,8,8,...].
print1("1, "); forstep(n=2,40,2,v=vector(n,i,if(i>1,8,4)); print1(contfracpnqn(v)[1,1],", ")) \\ Rick L. Shepherd, Jul 31 2006

vector(20,n,polchebyshev(n-1,1,33)) \\ Joerg Arndt, Jan 01 2021

a(n) = 66*a(n-1) - a(n-2), a(-1):= 33, a(0)=1.
a(n) = T(n, 33) = (S(n, 66)-S(n-2, 66))/2 = S(n, 66)-33*S(n-1, 66) with T(n, x), resp. S(n, x), Chebyshev polynomials of the first, resp.second, kind. See A053120 and A049310. S(n, 66) = A097316(n).
a(n) = ((33+8*sqrt(17))^n + (33-8*sqrt(17))^n)/2.
a(n) = Sum_{k=0..floor(n/2)} ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*33)^(n-2*k), for n>=1, a(0)=1.
G.f.: (1-33*x)/(1-66*x+x^2).

A-number for y values in Pell equation corrected by Wolfdieter Lang, Jun 28 2013

A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686

Offset: 0

Paul Barry, Nov 15 2003

The family of recurrences a(n) = 2*k*a(n-1) + a(n-2), a(0)=1, a(1)=k has solution a(n) = ((k+sqrt(k^2+1))^n + (k-sqrt(k^2+1))^n)/2; a(n) = Sum_{j=0..floor(n/2)} C(n,2k)*(k^2+1)^jk^(n-2j); a(n) = T(n,ki)*(-i)^n; e.g.f. exp(kx)*cosh(sqrt(k^2+1)*x).

Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (12,1).

Cf. A088320, A088317, A005667, A001077.

Essentially the same as A041060.

E.g.f.: exp(6x)*cosh(sqrt(37)x);
a(n) = ((6+sqrt(37))^n + (6-sqrt(37))^n)/2;
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*37^k*6^(n-2k).
a(n) = T(n, 6i)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.
G.f.: (1-6x)/(1-12*x-x^2). - Philippe Deléham, Nov 21 2008

cp's OEIS Frontend

A086594 a(n) = 8*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=8.

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A099370 Chebyshev polynomial of the first kind, T(n,x), evaluated at x=33.

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A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

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