A273627 - OEIS

%I #17 Sep 08 2022 08:46:16

%S 1,48,1715,58752,1998709,67914000,2307174311,78376578048,

%T 2662499775145,90446634986352,3072523201721819,104375342876112000,

%U 3545689138389464221,120449055384533383248,4091722194064948458575,138998105543576763850752,4721843866291934117429329

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

%C 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.

%H Colin Barker, <a href="/A273627/b273627.txt">Table of n, a(n) for n = 1..600</a>

%H P. Bala, <a href="/A273622/a273622.pdf">Lucas sequences and divisibility sequences</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (40,-206,40,-1).

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

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

%F a(n) = -a(-n).

%F a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4) for n>4.

%F O.g.f.: x*(x^2 + 8*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)).

%F 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

%p #A273627

%p A000129 := proc (n) option remember;

%p if n <= 1 then n else 2*A000129(n-1)+A000129(n-2) end if

%p end proc:

%p seq(1/8*(A000129(4*n)-2*A000129(2*n)), n = 1..20);

%t 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 *)

%t LinearRecurrence[{40, -206, 40, -1}, {1, 48, 1715, 58752}, 20] (* _Vincenzo Librandi_, Jun 02 2016 *)

%o (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

%Y Cf. A000129, A002203, A127595, A273626.

%K nonn,easy

%O 1,2

%A _Peter Bala_, May 30 2016