This site is supported by donations to The OEIS Foundation.

 Annual Appeal: Please make a donation to keep the OEIS running. In 2017 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A090248 a(n) = 27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27. 5
 2, 27, 727, 19602, 528527, 14250627, 384238402, 10360186227, 279340789727, 7531841136402, 203080369893127, 5475638145978027, 147639149571513602, 3980781400284889227, 107333458658120495527, 2894022602368968490002 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS a(n+1)/a(n) converges to ((27+sqrt(725))/2) = 26.96291201... Lim a(n)/a(n+1) as n approaches infinity = 0.03708798... = 2/(27+sqrt(725)) = (27-sqrt(725))/2. Lim a(n+1)/a(n) as n approaches infinity = 26.96291201... = (27+sqrt(725))/2 = 2/(27-sqrt(725)). Lim a(n)/a(n+1) = 27 - Lim a(n+1)/a(n). A Chebyshev T-sequence with Diophantine property. a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 29*(5*b)^2 =+4 with companion sequence b(n)=A097781(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 (27, -1). FORMULA a(n) = 27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27. a(n) = ((27+sqrt(725))/2)^n + ((27-sqrt(725))/2)^n, (a(n))^2 = a(2n)+2. a(n) = S(n, 27) - S(n-2, 27) = 2*T(n, 27/2) 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 := (27+5*sqrt(29))/2 and am := (27-5*sqrt(29))/2. G.f.: (2-27*x)/(1-27*x+x^2). a(-n) = a(n). - Michael Somos, Nov 01 2008 A087130(2*n) = a(n). - Michael Somos, Nov 01 2008 EXAMPLE a(4) = 528527 = 27a(3) - a(2) = 27*19602 - 727 = ((27+sqrt(725))/2)^4 + ((27-sqrt(725))/2)^4 = 528526.999998107 + 0.000001892 = 528527. (x;y) = (2;0), (27;1), (727;27), (19602;728), ... give the nonnegative integer solutions to x^2 - 29*(5*y)^2 = +4. MATHEMATICA a[0] = 2; a[1] = 27; a[n_] := 27a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *) RecurrenceTable[{a[0]==2, a[1]==27, a[n]==27a[n-1]-a[n-2]}, a, {n, 20}] (* or *) LinearRecurrence[{27, -1}, {2, 27}, 20] (* Harvey P. Dale, Jan 03 2018 *) PROG (Sage) [lucas_number2(n, 27, 1) for n in xrange(0, 16)] # Zerinvary Lajos, Jun 27 2008 (PARI) {a(n) = (-1)^n * subst(2 * poltchebi(2*n), 'x, -5/2 * I)}; /* Michael Somos, Nov 04 2008 */ CROSSREFS Cf. A046213, A078046. a(n)=sqrt(4 + 29*(5*A097781(n-1))^2), n>=1. Cf. A077428, A078355 (Pell +4 equations). Cf. A090733 for 2*T(n, 25/2). Cf. A087130. Sequence in context: A203429 A153850 A138458 * A300591 A251693 A182934 Adjacent sequences:  A090245 A090246 A090247 * A090249 A090250 A090251 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 10 13:02 EST 2018. Contains 318048 sequences. (Running on oeis4.)