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

1, 30, 995, 33660, 1142629, 38810970, 1318402631, 44786716920, 1521429030985, 51683794848150, 1755727563817259, 59643053188493940, 2026108079758297261, 68828031652259981010, 2338126968060165944975, 79427488882178225107440, 2698196495024745460575889

Offset

1,2

Comments

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.

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.

Links

Colin Barker, Table of n, a(n) for n = 1..600

P. Bala, Lucas sequences and divisibility sequences

Wikipedia, Lucas Sequence

Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Formula

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) ).

a(n) = (A082405(n) + A001109(n))/7 .

a(n) = 1/14*Pell(2*n)*A081555(n).

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 - 10*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)).

Maple

#A273626
A000129 := proc (n) option remember;
if n <= 1 then n else 2*A000129(n-1) + A000129(n-2) end if
end proc:
seq(1/14*(A000129(4*n) + A000129(2*n)), n = 1..20);

Mathematica

LinearRecurrence[{40, -206, 40, -1}, {1, 30, 995, 33660}, 100] (* G. C. Greubel, Jun 02 2016 *)

Prog

(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

Crossrefs

Cf. A000129, A001109, A029547, A081555, A127595, A215466, A273627.

Sequence in context: A276396 A291070 A004994 * A280958 A061466 A180812

Adjacent sequences: A273623 A273624 A273625 * A273627 A273628 A273629

Keyword

nonn,easy

Author

Peter Bala, May 31 2016

Status

approved