A238539 A fourth-order linear divisibility sequence: a(n) := (1/9)*(2^n + (-1)^n)*(2^(3*n) - (-1)^n).

1, 35, 399, 7735, 112871, 1893255, 29593159, 479082695, 7620584391, 122287263175, 1953732901319, 31282632909255, 500338874618311, 8006888009380295, 128098480026087879, 2049669505409577415, 32793961486615474631, 524709388585350492615, 8395302178969583120839

Offset

1,2

Comments

This is a divisibility sequence, that is, if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.

The sequence satisfies a homogeneous linear recurrence of the fourth order. However, it does not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer.

For sequences of a similar type see A238536 through A238541.

Links

Table of n, a(n) for n=1..19.

Peter Bala, A family of linear divisibility sequences of order four

Wikipedia, Divisibility sequence"

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for linear recurrences with constant coefficients, signature (7,138,112,-256).

Formula

a(n) = (1/9)*(2^n + (-1)^n)*(2^(3*n) - (-1)^n) = (1/9)*(4^n - 1)*(8^n - (-1)^n)/(2^n - (-1)^n).

O.g.f.: x*(1 + 28*x + 16*x^2)/((1 - x)*(1 + 2*x)*(1 + 8*x)*(1 - 16*x)).

Recurrence equation: a(n) = 7*a(n-1) + 138*a(n-2) + 112*a(n-4) - 256*a(n-4).

Maple

seq(1/9*(2^n + (-1)^n)*(2^(3*n) - (-1)^n), n = 1..20);

Crossrefs

Cf. A238536, A238537, A238538, A238540, A238541.

Sequence in context: A225697 A133458 A380581 * A075664 A133317 A322879

Adjacent sequences: A238536 A238537 A238538 * A238540 A238541 A238542

Keyword

nonn,easy

Author

Peter Bala, Mar 01 2014

Status

approved