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

Author

Peter Bala, Mar 01 2014

Keywords

• nonn
• easy

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

• 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).

Crossrefs

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

Programs

Maple

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

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).