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 A078365 A Chebyshev T-sequence with Diophantine property. 3
 2, 15, 223, 3330, 49727, 742575, 11088898, 165590895, 2472774527, 36926027010, 551417630623, 8234338432335, 122963658854402, 1836220544383695, 27420344506901023, 409468947059131650 (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 - 221*b^2 =+4 with companion sequence b(n)=A078364(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 (15,-1). FORMULA a(n)=15*a(n-1)-a(n-2), n >= 1; a(-1)=15, a(0)=2. a(n) = S(n, 15) - S(n-2, 15) = 2*T(n, 15/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 15)=A078364(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120. G.f.: (2-15*x)/(1-15*x+x^2). a(n) = ap^n + am^n, with ap := (15+sqrt(221))/2 and am := (15-sqrt(221))/2. a(n)=(15/2)*[15/2+(1/2)*sqrt(221)]^n-(1/2)*[15/2+(1/2)*sqrt(221)]^n*sqrt(221)+(1/2)*sqrt(221) *[15/2-(1/2)*sqrt(221)]^n+(15/2)*[15/2-(1/2)*sqrt(221)]^n, with n>=0 - Paolo P. Lava, Jun 19 2008 MATHEMATICA a[0] = 2; a[1] = 15; a[n_] := 15a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *) PROG (Sage) [lucas_number2(n, 15, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 26 2008 CROSSREFS a(n)=sqrt(4 + 221*A078364(n-1)^2), n>=1, (Pell equation d=221, +4). Cf. A077428, A078355 (Pell +4 equations). Sequence in context: A087962 A140054 A099085 * A207037 A218798 A176337 Adjacent sequences:  A078362 A078363 A078364 * A078366 A078367 A078368 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 29 2002 STATUS approved

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Last modified November 18 09:05 EST 2017. Contains 294865 sequences.