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A090731 a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23. 3
2, 23, 527, 12098, 277727, 6375623, 146361602, 3359941223, 77132286527, 1770682648898, 40648568638127, 933146396028023, 21421718540006402, 491766380024119223, 11289205022014735727, 259159949126314802498 (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 - 21*(5*b)^2 =+4 with companion sequence b(n)=A097778(n-1), n>=1; b(0):=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..733

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 (23,-1).

FORMULA

a(n) = S(n, 23) - S(n-2, 23) = 2*T(n, 23/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 23)=A097778(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

a(n) = ap^n + am^n, with ap := (23+5*sqrt(21))/2 and am := (23-5*sqrt(21))/2.

G.f.: (2-23*x)/(1-23*x+x^2).

EXAMPLE

(x;y) = (0;2), (23;1), (527;23), (12098;528), ... give the

nonnegative integer solutions to x^2 - 21*(5*y)^2 = 4.

MATHEMATICA

a[0] = 2; a[1] = 23; a[n_] := 23a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

LinearRecurrence[{23, -1}, {2, 23}, 30] (* Harvey P. Dale, Feb 20 2012 *)

PROG

(Sage) [lucas_number2(n, 23, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 26 2008

CROSSREFS

Cf. A037088, A051502.

a(n)=sqrt(4 + 21*(5*A097778(n-1))^2), n>=1.

Cf. A077428, A078355 (Pell +4 equations).

Sequence in context: A053066 A167417 A053161 * A090314 A084322 A073062

Adjacent sequences:  A090728 A090729 A090730 * A090732 A090733 A090734

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified May 22 20:08 EDT 2017. Contains 286906 sequences.