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

2, 27, 727, 19602, 528527, 14250627, 384238402, 10360186227, 279340789727, 7531841136402, 203080369893127, 5475638145978027, 147639149571513602, 3980781400284889227, 107333458658120495527, 2894022602368968490002, 78031276805304028734527

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004

Keywords

• easy
• nonn

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.

Examples

			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.
		

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 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.

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.

Programs

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 *)

PARI

{a(n) = (-1)^n * subst(2 * poltchebi(2*n), 'x, -5/2 * I)}; /* Michael Somos, Nov 04 2008 */

Python

def aupton(idx):
  alst = [2, 27]
  for n in range(2, idx+1): alst.append(27*alst[-1] - alst[-2])
  return alst
print(aupton(16)) # Michael S. Branicky, Feb 27 2021

Sage

[lucas_number2(n,27,1) for n in range(0,16)] # Zerinvary Lajos, Jun 27 2008

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

Extensions

More terms from Robert G. Wilson v, Jan 30 2004

Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004