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 A078369 A Chebyshev T-sequence with Diophantine property. 3
 2, 19, 359, 6802, 128879, 2441899, 46267202, 876634939, 16609796639, 314709501202, 5962870726199, 112979834296579, 2140653980908802, 40559445802970659, 768488816275533719, 14560728063432170002 (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 - 357*b^2 =+4 with companion sequence b(n)=A078368(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 (19,-1). FORMULA a(n)=19*a(n-1)-a(n-2), n >= 1; a(-1)=19, a(0)=2. a(n) = S(n, 19) - S(n-2, 19) = 2*T(n, 19/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 19)=A078368(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120. G.f.: (2-19*x)/(1-19*x+x^2). a(n) = ap^n + am^n, with ap := (19+sqrt(357))/2 and am := (19-sqrt(357))/2. MATHEMATICA a[0] = 2; a[1] = 19; a[n_] := 19a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *) PROG (Sage) [lucas_number2(n, 19, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 27 2008 CROSSREFS a(n)=sqrt(4 + 357*A078368(n-1)^2), n>=1, (Pell equation d=357, +4). Cf. A077428, A078355 (Pell +4 equations). Sequence in context: A137647 A233107 A187659 * A090308 A110818 A155927 Adjacent sequences:  A078366 A078367 A078368 * A078370 A078371 A078372 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 29 2002 STATUS approved

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Last modified December 12 00:07 EST 2018. Contains 318052 sequences. (Running on oeis4.)