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A090733 a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25. 3
2, 25, 623, 15550, 388127, 9687625, 241802498, 6035374825, 150642568127, 3760028828350, 93850078140623, 2342491924687225, 58468448039040002, 1459368709051312825, 36425749278243780623, 909184363247043202750 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

REFERENCES

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

LINKS

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)

FORMULA

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).

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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).

Sequence in context: A074209 A209467 A121252 * A197084 A119829 A059363

Adjacent sequences:  A090730 A090731 A090732 * A090734 A090735 A090736

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified November 21 13:53 EST 2017. Contains 295001 sequences.