A097725 -id:A097725 - cp's OEIS frontend

A029548 Expansion of 1/(1 - 32*x + x^2).

1, 32, 1023, 32704, 1045505, 33423456, 1068505087, 34158739328, 1092011153409, 34910198169760, 1116034330278911, 35678188370755392, 1140585993533893633, 36463073604713840864, 1165677769357309014015

Offset: 0

N. J. A. Sloane

From Bruno Berselli, Nov 21 2011: (Start)
A Diophantine property of these numbers: ((a(n+1) - a(n-1))/2)^2 - 255*a(n)^2 = 1.
More generally, for t(m) = m + sqrt(m^2-1) and u(n) = (t(m)^(n+1) - 1/t(m)^(n+1))/(t(m) - 1/t(m)), we can verify that ((u(n+1) - u(n-1))/2)^2 - (m^2-1)*u(n)^2 = 1. (End)
a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,31}. - Milan Janjic, Jan 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (32,-1).

Cf. A200441, A200442.

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), this sequence (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33), A097725 (m=51).

m:=17;; 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

I:=[1, 32]; [n le 2 select I[n] else 32*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

seq(simplify(ChebyshevU(n, 16)), n=0..20); # G. C. Greubel, Dec 22 2019

Table[GegenbauerC[n, 1, 16], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[1/(1-32x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
ChebyshevU[Range[0,20], 16] (* G. C. Greubel, Dec 22 2019 *)

vector( 21, n, polchebyshev(n-1, 2, 17) ) \\ G. C. Greubel, Dec 22 2019

[lucas_number1(n,32,1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009

[chebyshev_U(n,16) for n in (0..20)] # G. C. Greubel, Dec 22 2019

a(n) = 32*a(n-1) - a(n-2), a(-1)=0, a(0)=1.
a(n) = S(n, 32) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - Wolfdieter Lang, Nov 29 2002
a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap=16+sqrt(255) and am=16-sqrt(255).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*32^(n-2*k).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*31^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/15*(15 + sqrt(255)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/32*(15 + sqrt(255)). - Peter Bala, Dec 23 2012

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

Original entry on oeis.org

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

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

Original entry on oeis.org

1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623, 1279323224145592925115025, 130478425272533383961791927, 13307520054574259571177661529

Offset: 0

Wolfdieter Lang, Aug 31 2004

a(-1) = -1. - Artur Jasinski, Feb 10 2010

5*a(n) gives the x-values in the solution to the Pell equation x^2 - 26*y^2 = -1. - Colin Barker, Aug 24 2013

			(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).
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121. - Artur Jasinski, Feb 10 2010
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* Artur Jasinski, Feb 10 2010 *)
CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)
LinearRecurrence[{102,-1},{1,103},20] (* Harvey P. Dale, Aug 20 2017 *)

x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 102*x + x^2).
a(n) = S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), 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, 5*i)/(5*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=103. - Philippe Deléham, Nov 18 2008
a(n) = (1/5)*sinh((2*n-1)*arcsinh(5)), n >= 1. - Artur Jasinski, Feb 10 2010

More terms from Harvey P. Dale, Aug 20 2017

cp's OEIS Frontend

A029548 Expansion of 1/(1 - 32*x + x^2).

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

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

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