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 A078367 A Chebyshev T-sequence with Diophantine property. 4
 2, 17, 287, 4862, 82367, 1395377, 23639042, 400468337, 6784322687, 114933017342, 1947076972127, 32985375508817, 558804306677762, 9466687838013137, 160374888939545567, 2716906424134261502 (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 - 285*b^2 =+4 with companion sequence b(n)=A078366(n-1), n>=1. 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 Index entries for linear recurrences with constant coefficients, signature (17,-1). FORMULA a(n)=17*a(n-1)-a(n-2), n >= 1; a(-1)=17, a(0)=2. a(n) = S(n, 17) - S(n-2, 17) = 2*T(n, 17/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 17)=A078366(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120. G.f.: (2-17*x)/(1-17*x+x^2). a(n) = ap^n + am^n, with ap := (17+sqrt(285))/2 and am := (17-sqrt(285))/2. MATHEMATICA a[0] = 2; a[1] = 17; a[n_] := 17a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *) PROG (PARI) a(n)=if(n<0, 0, subst(2*poltchebi(n), x, 17/2)) (Sage) [lucas_number2(n, 17, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 26 2008 CROSSREFS a(n)=sqrt(4 + 285*A078366(n-1)^2), n>=1, (Pell equation d=285, +4). Cf. A077428, A078355 (Pell +4 equations). Sequence in context: A086534 A198287 A268705 * A090306 A304857 A007785 Adjacent sequences:  A078364 A078365 A078366 * A078368 A078369 A078370 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 29 2002 STATUS approved

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Last modified January 20 19:43 EST 2019. Contains 319335 sequences. (Running on oeis4.)