A238541 A fourth-order linear divisibility sequence: a(n) := A(n)/A(1) where A(n) := ( (3^n + 2^n)*(3^(3*n) - 2^(3*n)) ).

1, 91, 7063, 538447, 41441455, 3231753343, 254851186927, 20265345051679, 1621012954550479, 130194036583465855, 10485834936321976111, 846117830539227426271, 68360837263665964839823, 5527792975131721247371327, 447241733557623755497669615

Offset: 1

Author

Peter Bala, Mar 01 2014

Keywords

• nonn
• easy

Comments

The bivariate polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a sequence of divisibility polynomials in the polynomial ring Z[x,y]; that is, if n divides m then P(n,x,y) divides P(m,x,y) in Z[x,y] (see the Bala link). Here we consider the integer sequence coming from the normalized polynomials P(n,x,y)/P(1,x,y) when x = 3 and y = 2. Other cases include A238538(x = 2, y = 1), A238539(x = 2, y = -1) and A238540(x = 3, y = 1). See also A238536, A238537 and A215466.

Links

• Peter Bala, A family of linear divisibility sequences of order four
• Wikipedia, Divisibility sequence
• Index entries for linear recurrences with constant coefficients, signature (175,-10158,226800,-1679616).

Crossrefs

Cf. A215466, A238536, A238537, A238538, A238539, A238540.

Programs

Maple

#A238541
seq(1/95*(3^n + 2^n)*(3^(3*n) - 2^(2*n)), n = 1..20);

Mathematica

LinearRecurrence[{175,-10158,226800,-1679616},{1,91,7063,538447},20] (* Harvey P. Dale, Apr 12 2018 *)

Formula

a(n) = (1/95)*(3^n + 2^n)*(3^(3*n) - 2^(3*n)).

a(n) = (1/95)*(9^n - 4^n)*(27^n - 8^n)/(3^n - 2^n).

O.g.f.: x*(1 - 84*x + 1296*x^2)/((1 - 16*x)*(1 - 24*x)*(1 - 54*x)*(1 - 81*x)).

Recurrence equation: a(n) = 175*a(n-1) - 10158*a(n-2) + 226800*a(n-4) - 1679616*a(n-4).