A097780 -id:A097780 - cp's OEIS frontend

A154022 a(n) = 5*A097780(n-2).

0, 5, 125, 3120, 77875, 1943755, 48516000, 1210956245, 30225390125, 754423796880, 18830369531875, 470004814499995, 11731289992968000, 292812245009700005, 7308574835249532125, 182421558636228603120

Offset: 1

Vincenzo Librandi, Jan 04 2009

If a(n)=x and a(n+1)=y, then (x^2+y^2)/(xy+1)=25.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (25,-1).

Cf. A065100, A154021-A154027.

I:=[0, 5]; [n le 2 select I[n] else 25*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 29 2012

CoefficientList[Series[(5*z)/(z^2-25*z+1),{z,0,25}],z] (* Vincenzo Librandi, Jan 29 2012 *)
LinearRecurrence[{25,-1},{0,5},20] (* Harvey P. Dale, Mar 06 2018 *)

concat(0,Vec(5/(1-25*x+x^2)+O(x^98))) \\ Charles R Greathouse IV, Dec 27 2011

a(n) = +25*a(n-1) -a(n-2).

G.f.: 5*x^2/(1 -25*x +x^2). - R. J. Mathar, Jan 05 2011

Edited by N. J. A. Sloane, Jun 23 2010 at the suggestion of Joerg Arndt.

A090733 a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.

Original entry on oeis.org

2, 25, 623, 15550, 388127, 9687625, 241802498, 6035374825, 150642568127, 3760028828350, 93850078140623, 2342491924687225, 58468448039040002, 1459368709051312825, 36425749278243780623, 909184363247043202750

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 69*(3*b)^2 =+4 together with the companion sequence b(n)=A097780(n-1), n>=0.

			(x,y) =(2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Indranil Ghosh, Table of n, a(n) for n = 0..714
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 (25,-1)

Cf. A046069, A082974.
a(n)=sqrt(4 + 69*(3*A097780(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A097779 for 2*T(n, 23/2).

a[0] = 2; a[1] = 25; a[n_] := 25a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

[lucas_number2(n,25,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = S(n, 25) - S(n-2, 25) = 2*T(n, 25/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 25)=A097780(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.
G.f.: (2-25*x)/(1-25*x+x^2).

Extension, Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

cp's OEIS Frontend

A154022 a(n) = 5*A097780(n-2).

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