OFFSET
1,2
COMMENTS
The sequence of Pell numbers A000129 is a well-known divisibility sequence of order 2, that is, the sequence satisfies a linear recurrence of order 2 and Pell(n) divides Pell(m) whenever n divides m. The linear combinations Pell(4*n) - 2*Pell(2*n) and Pell(4*n) + Pell(2*n) are also divisibility sequences, this time of order 4. Cf. A127595 and A273626.
LINKS
Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).
FORMULA
a(n) = -a(-n).
a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4) for n>4.
O.g.f.: x*(x^2 + 8*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)).
a(n) = ((17+12*sqrt(2))^(1-n)*(-1+2*(3+2*sqrt(2))^n+(17+12*sqrt(2))^(2*n)-2*(99+70*sqrt(2))^n))/(384+272*sqrt(2)). - Colin Barker, Jun 02 2016
MAPLE
MATHEMATICA
CoefficientList[Series[(x^2 + 8*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{40, -206, 40, -1}, {1, 48, 1715, 58752}, 20] (* Vincenzo Librandi, Jun 02 2016 *)
PROG
(Magma) I:=[1, 48, 1715, 58752]; [n le 4 select I[n] else 40*Self(n-1)-206*Self(n-2)+40*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 02 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, May 30 2016
STATUS
approved