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A078363 A Chebyshev T-sequence with Diophantine property. 4
2, 13, 167, 2158, 27887, 360373, 4656962, 60180133, 777684767, 10049721838, 129868699127, 1678243366813, 21687295069442, 280256592535933, 3621648407897687, 46801172710133998, 604793596823844287, 7815515585999841733, 100996909021174098242 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 = +4 with companion sequence b(n)=A078362(n-1), n>=1.
Except for the first term, positive values of x (or y) satisfying x^2 - 13xy + y^2 + 165 = 0. - Colin Barker, Feb 26 2014
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) = 13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2.
a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-13*x)/(1-13*x+x^2).
a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.
a(n) = sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).
E.g.f.: 2*exp(13*x/2)*cosh(sqrt(165)*x/2). - Stefano Spezia, Sep 24 2022
MATHEMATICA
a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{13, -1}, {2, 13}, 20] (* Harvey P. Dale, Oct 28 2016 *)
PROG
(PARI) a(n)=if(n<0, 0, 2*subst(poltchebi(n), x, 13/2))
(PARI) a(n)=if(n<0, 0, polsym(1-13*x+x^2, n)[n+1])
(PARI) Vec((2-13*x)/(1-13*x+x^2) + O(x^100)) \\ Colin Barker, Feb 26 2014
(Sage) [lucas_number2(n, 13, 1) for n in range(0, 20)] - Zerinvary Lajos, Jun 25 2008
CROSSREFS
Cf. A078362.
Cf. A077428, A078355 (Pell +4 equations).
Sequence in context: A177448 A258224 A351021 * A143851 A088316 A006905
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
EXTENSIONS
More terms from Colin Barker, Feb 26 2014
STATUS
approved

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Last modified February 28 04:16 EST 2024. Contains 370379 sequences. (Running on oeis4.)