A238540 A fourth-order linear divisibility sequence: a(n) := (3^n + 1)*(3^(3*n) - 1)/( (3 + 1)*(3^3 - 1)).

1, 70, 5299, 419020, 33664741, 2719393810, 220069738519, 17820217484440, 1443290970139081, 116902609136432350, 9469004435040169339, 766986472802959676260, 62125826363286791503021, 5032189831214900660779690, 407607319514701058318401759, 33016191346720726553176114480

Offset

1,2

Comments

This is a divisibility sequence, that is, if n | m then a(n) | a(m). More generally, the polynomials P(n,x) := (x^n + 1)*(x^(3*n) - 1) form a sequence of divisibility polynomials in the polynomial ring Z[x]; that is, if n divides m then P(n,x) divides P(m,x) in Z[x]. See the Bala link for a proof and generalization. Here we consider the integer sequence coming from the normalized polynomials P(n,x)/P(n,1) at x = 3.

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 discovered 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 parameter.

For sequences of a similar type see A238536 through A238541.

Links

Table of n, a(n) for n=1..16.

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 (112,-2622,9072,-6561).

Formula

a(n) = (1/104)*(3^n + 1)*(3^(3*n) - 1) = (1/104)*(9^n - 1)*(27^n - 1)/(3^n - 1).

O.g.f.: x*(1 - 42*x + 81*x^2)/((1 - x)*(1 - 3*x)*(1 - 27*x)*(1 - 81*x)).

Recurrence equation: a(n) = 112*a(n-1) - 2622*a(n-2) + 9072*a(n-3) - 6561*a(n-4).

Maple

#A238540
seq(1/104*(3^n + 1)*(3^(3*n) - 1), n = 1..20);

Mathematica

LinearRecurrence[{112, -2622, 9072, -6561}, {1, 70, 5299, 419020}, 16] (* Jean-François Alcover, Nov 14 2019 *)

Crossrefs

Cf. A238536, A238537, A238538, A238539, A238541.

Sequence in context: A363297 A146349 A180884 * A060057 A363296 A049216

Adjacent sequences: A238537 A238538 A238539 * A238541 A238542 A238543

Keyword

nonn,easy

Author

Peter Bala, Mar 01 2014

Status

approved