A078989 -id:A078989 - cp's OEIS frontend

A097316 Chebyshev U(n,x) polynomial evaluated at x=33.

1, 66, 4355, 287364, 18961669, 1251182790, 82559102471, 5447649580296, 359462313197065, 23719065021425994, 1565098829100918539, 103272803655639197580, 6814439942443086121741, 449649763397588044837326

Offset: 0

Wolfdieter Lang, Aug 31 2004

Used to form integer solutions of Pell equation a^2 - 17*b^2 =-1. See A078989 with A078988.

Indranil Ghosh, Table of n, a(n) for n = 0..548
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (66,-1).

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), this sequence (m=33).

m:=33;; a:=[1,2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

m:=33; I:=[1, 2*m]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 22 2019

seq( simplify(ChebyshevU(n, 33)), n=0..20); # G. C. Greubel, Dec 22 2019

LinearRecurrence[{66, -1},{1, 66},14] (* Ray Chandler, Aug 11 2015 *)
ChebyshevU[Range[21] -1, 33] (* G. C. Greubel, Dec 22 2019 *)

vector( 21, n, polchebyshev(n-1, 2, 33) ) \\ G. C. Greubel, Dec 22 2019

[chebyshev_U(n,33) for n in (0..20)] # G. C. Greubel, Dec 22 2019

a(n) = 66*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.
a(n) = S(n, 66) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.
G.f.: 1/(1-66*x+x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*66^(n-2*k).
a(n) = ((33+8*sqrt(17))^(n+1) - (33-8*sqrt(17))^(n+1))/(16*sqrt(17)).

A097775 Pell equation solutions (14a(n))^2 - 197b(n)^2 = -1 with b(n) = A097776(n), n >= 0.

Original entry on oeis.org

1, 787, 618581, 486203879, 382155630313, 300373839222139, 236093455472970941, 185569155627915937487, 145857120230086453893841, 114643510931692324844621539, 90109653735189937241418635813, 70826073192348358979430203127479, 55669203419532074967894898239562681

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (14*1=14;1), (11018=14*787;785), (8660134=14*618581;617009), ... give the positive integer solutions to x^2 - 197*y^2 =-1.

Colin Barker, Table of n, a(n) for n = 0..345
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 (786,-1).

Cf. A097774 for S(n, 2*393).

Cf. similar sequences of the type (1/k)*sinh((2*n + 1)*arcsinh(k)): A002315 (k=1), A049629 (k=2), A097314 (k=3), A078989 (k=4), A097726 (k=5), A097729 (k=6), A097732 (k=7), A097735 (k=8), A097738 (k=9), A097741 (k=10), A097766 (k=11), A097769 (k=12), A097772 (k=13), this sequence (k=14).

LinearRecurrence[{786, -1}, {1, 787}, 20] (* Harvey P. Dale, Dec 12 2017 *)

Vec((1+x)/(1-2*393*x+x^2) + O(x^100)) \\ Colin Barker, Apr 04 2015

G.f.: (1 + x)/(1 - 2*393*x + x^2).
a(n) = S(n, 2*393) + S(n-1, 2*393) = S(2*n, 2*sqrt(197)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) = 0 = U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 14*i)/(14*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 786*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=787. - Philippe Deléham, Nov 18 2008
a(n) = (1/14)*sinh((2*n + 1)*arcsinh(14)). - Bruno Berselli, Apr 05 2018

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

A041024 Numerators of continued fraction convergents to sqrt(17).

Original entry on oeis.org

4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236

Offset: 0

N. J. A. Sloane

a(2*n+1) with b(2*n+1) := A041025(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 17*b^2 = +1, a(2*n) with b(2*n) := A041025(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 17*b^2 = -1 (cf. Emerson reference).

Bisection: a(2*n) = 4*S(2*n,2*sqrt(17)) = 4*A078989(n), n >= 0 and a(2*n+1) = T(n+1,33), n >= 0, with S(n,x), resp. T(n,x), Chebyshev's polynomials of the second, resp. first kind. See A049310, resp. A053120. - Wolfdieter Lang, Jan 10 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,1).

Cf. A010473, A040012, A041025.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[17],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011*)
LinearRecurrence[{8, 1}, {4, 33}, 25] (* Sture Sjöstedt, Dec 07 2011 *)
CoefficientList[Series[(4 + x)/(1 - 8 x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

G.f.: (4+x)/(1-8*x-x^2).
a(n) = 4*A041025(n) + A041025(n-1).
a(n) = ((-i)^(n+1))*T(n+1, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.
a(n) = 8*a(n-1) + a(n-2), n > 1. - Philippe Deléham, Nov 20 2008
a(n) = ((4 + sqrt(17))^n + (4 - sqrt(17))^n)/2. - Sture Sjöstedt, Dec 08 2011

A195619 Denominators of Pythagorean approximations to 4.

Original entry on oeis.org

16, 1040, 68640, 4529184, 298857520, 19720067120, 1301225572416, 85861167712320, 5665535843440720, 373839504499375184, 24667741761115321440, 1627697116729111839840, 107403341962360266108016, 7086992872399048451289200

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for a discussion and references.

Colin Barker, Table of n, a(n) for n = 1..549
Index entries for linear recurrences with constant coefficients, signature (65,65,-1).

Cf. A078988, A078989, A195500, A195620.

I:=[16, 1040, 68640]; [n le 3 select I[n] else 65*Self(n-1) +65*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 13 2023

r = 4; z = 20;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195619, A195620 *)
Sqrt[a^2 + b^2] (* A078988 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(LucasL[2*n+1,8] - 8*(-1)^n)/34, {n,40}] (* G. C. Greubel, Feb 13 2023 *)
LinearRecurrence[{65,65,-1},{16,1040,68640},20] (* Harvey P. Dale, May 01 2023 *)

Vec(16*x/((x+1)*(x^2-66*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015

A078989=BinaryRecurrenceSequence(66,-1,1,67)
[4*(A078989(n) - (-1)^n)/17 for n in range(1,41)] # G. C. Greubel, Feb 13 2023

From Colin Barker, Jun 03 2015: (Start)
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
G.f.: 16*x/((1+x)*(1-66*x+x^2)). (End)
a(n) = ((4+sqrt(17))^(2*n+1) + (4-sqrt(17))^(2*n+1) - 8*(-1)^n)/34. - Colin Barker, Mar 03 2016
a(n) = 4*(A078989(n) - (-1)^n)/17. - G. C. Greubel, Feb 13 2023

A195620 Numerators of Pythagorean approximations to 4.

Original entry on oeis.org

63, 4161, 274559, 18116737, 1195430079, 78880268481, 5204902289663, 343444670849281, 22662143373762879, 1495358017997500737, 98670967044461285759, 6510788466916447359361, 429613367849441064432063, 28347971489596193805156801

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for discussion and list of related sequences; see A195616 for Mathematica program.

Colin Barker, Table of n, a(n) for n = 1..549
Index entries for linear recurrences with constant coefficients, signature (65,65,-1).

Cf. A078989, A195500, A195619, A195620.

I:=[63,4161,274559]; [n le 3 select I[n] else 65*Self(n-1) +65*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 15 2023

LinearRecurrence[{65,65,-1}, {63,4161,274559}, 40] (* G. C. Greubel, Feb 15 2023 *)

Vec(x*(63+66*x-x^2)/((1+x)*(1-66*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 03 2015

A078989=BinaryRecurrenceSequence(66, -1, 1, 67)
[(16*A078989(n) + (-1)^n)/17 for n in range(1, 41)] # G. C. Greubel, Feb 15 2023

From Colin Barker, Jun 03 2015: (Start)
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
G.f.: x*(63+66*x-x^2) / ((1+x)*(1-66*x+x^2)). (End)
a(n) = ((-1)^n - 2*(-4+sqrt(17))*(33+8*sqrt(17))^(-n) + 2*(4+sqrt(17))*(33+8*sqrt(17))^n)/17. - Colin Barker, Mar 03 2016
a(n) = (1/17)*(A078989(n) + (-1)^n) - [n=0]. - G. C. Greubel, Feb 15 2023

A004298 Expansion of (1+2x+x^2)/(1-66x+x^2).

Original entry on oeis.org

1, 68, 4488, 296140, 19540752, 1289393492, 85080429720, 5614018968028, 370440171460128, 24443437297400420, 1612896421456967592, 106426720378862460652, 7022550648583465435440, 463381916086129856278388, 30576183911035987048938168, 2017564756212289015373640700

Offset: 0

N. J. A. Sloane

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
J. M. Alonso, Growth functions of amalgams, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (66,-1).

Pairwise sums of A078989.

Cf. A097316.

CoefficientList[Series[(1+2*x+x^2)/(1-66*x+x^2),{x,0,50}],x] (* Vincenzo Librandi, Feb 25 2012 *)
LinearRecurrence[{66,-1},{1,68,4488},20] (* Harvey P. Dale, Sep 23 2020 *)

Vec((1+2*x+x^2)/(1-66*x+x^2) + O(x^50)) \\ Colin Barker, Apr 16 2016

For n > 0, a(n) = 68*A097316(n-1). - Gerald McGarvey, Jun 16 2007
From Colin Barker, Apr 16 2016: (Start)
a(n) = (sqrt(17)*(33+8*sqrt(17))^(-n)*(-1+(33+8*sqrt(17))^(2*n)))/4 for n>0.
a(n) = 66*a(n-1) - a(n-2) for n>2.
(End)
a(n) = (-(-1)^(2^n) + sqrt(17)*sinh(n*log(33+8*sqrt(17))) + 1)/2. - Ilya Gutkovskiy, Apr 16 2016

cp's OEIS Frontend

A097316 Chebyshev U(n,x) polynomial evaluated at x=33.

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A097775 Pell equation solutions (14*a(n))^2 - 197*b(n)^2 = -1 with b(n) = A097776(n), n >= 0.

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A078988 Chebyshev sequence with Diophantine property.

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A041024 Numerators of continued fraction convergents to sqrt(17).

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A195619 Denominators of Pythagorean approximations to 4.

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A195620 Numerators of Pythagorean approximations to 4.

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Mathematica

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A004298 Expansion of (1+2*x+x^2)/(1-66*x+x^2).

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A097775 Pell equation solutions (14a(n))^2 - 197b(n)^2 = -1 with b(n) = A097776(n), n >= 0.

A004298 Expansion of (1+2x+x^2)/(1-66x+x^2).