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 A090251 a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29. 2
 2, 29, 839, 24302, 703919, 20389349, 590587202, 17106639509, 495501958559, 14352450158702, 415725552643799, 12041688576511469, 348793243166188802, 10102962363242963789, 292637115290879761079, 8476373381072270107502 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS a(n+1)/a(n) converges to ((29+sqrt(837))/2) =28.9654761... Lim a(n)/a(n+1) as n approaches infinity = 0.0345238... =2/(29+sqrt(837)) =(29-sqrt(837))/2. Lim a(n+1)/a(n) as n approaches infinity = 28.9654761... = (29+sqrt(837))/2=2/(29-sqrt(837)). Lim a(n)/a(n+1) = 29 - Lim a(n+1)/a(n). A Chebyshev T-sequence with a Diophantine property. a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 93*(3*b)^2 =+4 with companion sequence b(n)=A097782(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 Index entries for linear recurrences with constant coefficients, signature (29, -1). FORMULA a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29. a(n) = ((29+sqrt(837))/2)^n + ((29-sqrt(837))/2)^n, (a(n))^2 =a(2n)+2. a(n) = S(n, 29) - S(n-2, 29) = 2*T(n, 29/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 27)=A097781(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 := (29+3*sqrt(93))/2 and am := (29-3*sqrt(93))/2. G.f.: (2-29*x)/(1-29*x+x^2). EXAMPLE a(4) =703919 = 29a(3) - a(2) = 29*24302 - 839= ((29+sqrt(837))/2)^4 + ((29-sqrt(837))/2)^4 = 703918.99999857 + 0.00000142 =703919. (x,y) = (2;0), (29;1), (839;29), (24302,840), ..., give the nonnegative integer solutions to x^2 - 93*(3*y)^2 =+4. MATHEMATICA a[0] = 2; a[1] = 29; a[n_] := 29a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *) LinearRecurrence[{29, -1}, {2, 29}, 30] (* Harvey P. Dale, May 28 2013 *) PROG (Sage) [lucas_number2(n, 29, 1) for n in range(0, 16)] # Zerinvary Lajos, Jun 27 2008 CROSSREFS Cf. A083148, A007610. a(n)=sqrt(4 + 93*(3*A097782(n-1))^2), n>=1. Cf. A077428, A078355 (Pell +4 equations). Cf. A090248 for 2*T(n, 27/2). Sequence in context: A006988 A282735 A245252 * A087281 A024234 A077282 Adjacent sequences:  A090248 A090249 A090250 * A090252 A090253 A090254 KEYWORD easy,nonn AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004 EXTENSIONS More terms from Robert G. Wilson v, Jan 30 2004 Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified September 21 11:17 EDT 2020. Contains 337268 sequences. (Running on oeis4.)