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A090248 a(n) = 27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27. 4
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

Table of n, a(n) for n=0..15.

Tanya Khovanova, Recursive Sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for linear recurrences with constant coefficients, signature (27, -1).

Index entries for sequences related to Chebyshev polynomials.

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 *)

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 * A251693 A182934 A078102

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

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Last modified August 21 21:34 EDT 2017. Contains 290908 sequences.