A273627 A divisibility sequence: (1/8)*(Pell(4*n) - 2*Pell(2*n)).

1, 48, 1715, 58752, 1998709, 67914000, 2307174311, 78376578048, 2662499775145, 90446634986352, 3072523201721819, 104375342876112000, 3545689138389464221, 120449055384533383248, 4091722194064948458575, 138998105543576763850752, 4721843866291934117429329

Offset: 1

Author

Peter Bala, May 30 2016

Keywords

• nonn
• easy

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
• P. Bala, Lucas sequences and divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Crossrefs

Cf. A000129, A002203, A127595, A273626.

Programs

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

Maple

#A273627
A000129 := proc (n) option remember;
if n <= 1 then n else 2*A000129(n-1)+A000129(n-2) end if
end proc:
seq(1/8*(A000129(4*n)-2*A000129(2*n)), n = 1..20);

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 *)

Formula

a(n) = A000129(4*n) - 2*A000129(2*n).

a(n) = A000129(2*n)*(A002203(2*n) - 2).

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