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A273626 A fourth-order divisibility sequence: a(n) = (1/14)*(Pell(4*n) + Pell(2*n)).

%I A273626 #19 Sep 08 2022 08:46:16
%S A273626 1,30,995,33660,1142629,38810970,1318402631,44786716920,1521429030985,
%T A273626 51683794848150,1755727563817259,59643053188493940,
%U A273626 2026108079758297261,68828031652259981010,2338126968060165944975,79427488882178225107440,2698196495024745460575889
%N A273626 A fourth-order divisibility sequence: a(n) = (1/14)*(Pell(4*n) + Pell(2*n)).
%C A273626 This is a divisibility sequence, that is, a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4.
%C A273626 A000129(n) = Pell(n) is the Lucas sequence U_n(2,-1). In general, if U(n) = U_n(P,Q) is the Lucas sequence with integer parameters P and Q then when Q = 1 or Q = -1 both U(4*n) + U(2*n) and U(4*n) - 2*U(2*n) are divisibility sequences of the fourth order. Cf. A127595, A215466 and A273627.
%H A273626 Colin Barker, <a href="/A273626/b273626.txt">Table of n, a(n) for n = 1..600</a>
%H A273626 P. Bala, <a href="/A273622/a273622.pdf">Lucas sequences and divisibility sequences</a>
%H A273626 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas_sequence">Lucas Sequence</a>
%H A273626 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (40,-206,40,-1).
%F A273626 a(n) = sqrt(2)/56*( (sqrt(2) + 1)^(4*n) - (sqrt(2) - 1)^(4*n) + (sqrt(2) + 1)^(2*n) - (sqrt(2) - 1)^(2*n) ).
%F A273626 a(n) = (A082405(n) + A001109(n))/7 .
%F A273626 a(n) = 1/14*Pell(2*n)*A081555(n).
%F A273626 a(n) = -a(-n).
%F A273626 a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4) for n>4.
%F A273626 O.g.f.: x*(x^2 - 10*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)).
%p A273626 #A273626
%p A273626 A000129 := proc (n) option remember;
%p A273626 if n <= 1 then n else 2*A000129(n-1) + A000129(n-2) end if
%p A273626 end proc:
%p A273626 seq(1/14*(A000129(4*n) + A000129(2*n)), n = 1..20);
%t A273626 LinearRecurrence[{40,-206,40,-1},{1,30,995,33660},100] (* _G. C. Greubel_, Jun 02 2016 *)
%o A273626 (Magma) I:=[1,30,995,33660]; [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 04 2016
%Y A273626 Cf. A000129, A001109, A029547, A081555, A127595, A215466, A273627.
%K A273626 nonn,easy
%O A273626 1,2
%A A273626 _Peter Bala_, May 31 2016