A097726 -id:A097726 - cp's OEIS frontend

A097727 Pell equation solutions (5b(n))^2 - 26a(n)^2 = -1 with b(n)=A097726(n), n >= 0.

1, 101, 10301, 1050601, 107151001, 10928351501, 1114584702101, 113676711262801, 11593909964103601, 1182465139627304501, 120599850332020955501, 12300002268726510156601, 1254479631559772015017801, 127944622416828019021659101, 13049097006884898168194210501

Offset: 0

Wolfdieter Lang, Aug 31 2004

Hypotenuses of primitive Pythagorean triples in A195622 and A195623. - Clark Kimberling, Sep 22 2011

			(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 =-1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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 (102,-1).

Cf. A097725 for S(n, 102).

Row 5 of array A188647.

a:=[1,101];; for n in [3..20] do a[n]:=102*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,101]; [n le 2 select I[n] else 102*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

LinearRecurrence[{102,-1},{1,101},20] (* Harvey P. Dale, Apr 12 2014 *)
CoefficientList[Series[(1-x)/(1-102x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Apr 13 2014 *)

my(x='x+O('x^20)); Vec((1-x)/(1-102*x+x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-102*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = S(n, 2*51) - S(n-1, 2*51) = T(2*n+1, sqrt(26))/sqrt(26), 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, 10*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-102*x+x^2).
a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=101. - Philippe Deléham, Nov 18 2008

More terms from Harvey P. Dale, Apr 12 2014

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

A097725 Chebyshev U(n,x) polynomial evaluated at x=51.

Original entry on oeis.org

1, 102, 10403, 1061004, 108212005, 11036563506, 1125621265607, 114802332528408, 11708712296632009, 1194173851923936510, 121794024183944892011, 12421796292910455048612, 1266901427852682470066413, 129211523844680701491725514, 13178308530729578869685936015

Offset: 0

Wolfdieter Lang, Aug 31 2004

Used to form integer solutions of Pell equation a^2 - 26*b^2 =-1. See A097726 with A097727.

Indranil Ghosh, Table of n, a(n) for n = 0..496
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 sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (102,-1).

ChebyshevU[Range[0,20],51] (* Harvey P. Dale, Oct 08 2012 *)
LinearRecurrence[{102, -1},{1, 102},15] (* Ray Chandler, Aug 11 2015 *)

a(n) = 102*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.
a(n) = S(n, 2*51)= U(n, 51), Chebyshev's polynomials of the second kind. See A049310.
G.f.: 1/(1-102*x+x^2).
a(n)= sum((-1)^k*binomial(n-k, k)*102^(n-2*k), k=0..floor(n/2)), n>=0.
a(n) = ((51+10*sqrt(26))^(n+1) - (51-10*sqrt(26))^(n+1))/(20*sqrt(26)).

More terms from Harvey P. Dale, Oct 08 2012

A195622 Denominators of Pythagorean approximations to 5.

Original entry on oeis.org

20, 2020, 206040, 21014040, 2143226060, 218588044060, 22293837268080, 2273752813300080, 231900493119340100, 23651576545359390100, 2412228907133538450120, 246023696951075562522120, 25092004860102573838806140, 2559138472033511455995704140

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for a discussion and references.

Colin Barker, Table of n, a(n) for n = 1..498
Index entries for linear recurrences with constant coefficients, signature (101,101,-1).

Cf. A097726, A195500, A195623.

I:=[20,2020,206040]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 15 2023

r = 5; 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}]]  (* A195622, A195623 *)
Sqrt[a^2 + b^2] (* A097727 *)
(* by Peter J. C. Moses, Sep 02 2011 *)
LinearRecurrence[{101,101,-1},{20,2020,206040},20] (* Harvey P. Dale, Oct 17 2021 *)

Vec(20*x/((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015

A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
[(5/26)*(A097726(n) - (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 15 2023

From Colin Barker, Jun 03 2015: (Start)
a(n) = 101*a(n-1) + 101*a(n-2) - a(n-3).
G.f.: 20*x/((1+x)*(1-102*x+x^2)). (End)
a(n) = (5/26)*(A097726(n) - (-1)^n). - G. C. Greubel, Feb 15 2023

A195623 Numerators of Pythagorean approximations to 5.

Original entry on oeis.org

99, 10101, 1030199, 105070201, 10716130299, 1092940220301, 111469186340399, 11368764066500401, 1159502465596700499, 118257882726796950501, 12061144535667692250599, 1230118484755377812610601, 125460024300512869194030699, 12795692360167557279978520701, 1305035160712790329688615080799

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for discussion and list of related sequences; see A195622 for Mathematica program.

Colin Barker, Table of n, a(n) for n = 1..497
Index entries for linear recurrences with constant coefficients, signature (101,101,-1).

Cf. A097726, A097727, A195500, A195622.

I:=[99,10101,1030199]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 16 2023

Table[(5*LucasL[2*n+1,10] +2*(-1)^n)/52, {n,40}] (* G. C. Greubel, Feb 16 2023 *)

Vec(-x*(x^2-102*x-99) / ((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015

A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
[(1/26)*(25*A097726(n) + (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 16 2023

From Colin Barker, Jun 03 2015: (Start)
a(n) = 101*a(n-1) + 101*a(n-2) - a(n-3).
G.f.: x*(99+102*x-x^2)/((1+x)*(1-102*x+x^2)). (End)
a(n) = (1/26)*(25*A097726(n) + (-1)^n). - G. C. Greubel, Feb 16 2023
E.g.f.: (5*exp(51*x)*(5*cosh(10*sqrt(26)*x) + sqrt(26)*sinh(10*sqrt(26)*x)) + exp(-x) - 26)/26. - Stefano Spezia, Aug 05 2024

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A097727 Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n)=A097726(n), n >= 0.

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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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A097725 Chebyshev U(n,x) polynomial evaluated at x=51.

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A195622 Denominators of Pythagorean approximations to 5.

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A195623 Numerators of Pythagorean approximations to 5.

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A097727 Pell equation solutions (5b(n))^2 - 26a(n)^2 = -1 with b(n)=A097726(n), n >= 0.

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