A266507 a(n) = 6*a(n - 1) - a(n - 2) with a(0) = 2, a(1) = 8.

2, 8, 46, 268, 1562, 9104, 53062, 309268, 1802546, 10506008, 61233502, 356895004, 2080136522, 12123924128, 70663408246, 411856525348, 2400475743842, 13990997937704, 81545511882382, 475282073356588, 2770146928257146, 16145599496186288, 94103450048860582, 548475100796977204

Offset

0,1

Comments

Bisection of A078343 = A078343(2*n + 1).

Quadrisection of A266504 = A266504(4*n + 1).

Octasection of A266506 = A266506(8*n + 2).

References

Edward J. Barbeau, Pell's Equation, New York: Springer-Verlag, 2003, p. 52, Exercise 4.9.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-1).

Formula

a(n) = (-sqrt(2)*(1+sqrt(2))^(2*n+1) - 3 *(1-sqrt(2))^(2*n+1) - sqrt(2)*(1-sqrt(2))^(2*n+1) + 3*(1+sqrt(2))^(2*n+1))/sqrt(8).

a(n) = 2 * A038723(n).

G.f.: 2*(1-2*x) / (1-6*x+x^2). - Colin Barker, Dec 31 2015

E.g.f.: exp(3*x)*(4*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x))/2. - Stefano Spezia, Nov 30 2025

Mathematica

LinearRecurrence[{6, -1}, {2, 8}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[2 (1 - 2 x)/(1 - 6 x + x^2), {x, 0, n}], {n, 0, 22}] (* Michael De Vlieger, Dec 31 2015 *)

Prog

(Magma) I:=[2, 8]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015
(PARI) Vec(2*(1-2*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Dec 31 2015

Crossrefs

Bisection of A078343 = A078343(2n + 1).

Quadrisection of A266504 = A266504(4n + 1).

Octasection of A266506 = A266506(8n + 2).

Equals 2*A038723.

Sequence in context: A119501 A183277 A269006 * A202081 A258315 A334498

Adjacent sequences: A266504 A266505 A266506 * A266508 A266509 A266510

Keyword

nonn,easy

Author

Raphie Frank, Dec 30 2015

Status

approved