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A090733 a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25. 4
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
Tanya Khovanova, Recursive Sequences
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 range(0, 20)] # Zerinvary Lajos, Jun 26 2008
CROSSREFS
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: A209467 A121252 A322727 * A330767 A197084 A366038
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 March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)