A238600 A sixth-order linear divisibility sequence related to the Fibonacci numbers: a(n) := (1/6)*Fibonacci(3*n)*Fibonacci(4*n)/Fibonacci(n).

1, 28, 408, 7896, 137555, 2496144, 44599477, 801617712, 14375440584, 258018516140, 4629531440711, 83076469908768, 1490726895438793, 26750144944686436, 480010941060482040, 8613453244178393184, 154562103244937408987, 2773504708179098411952

Offset: 1

Author

Peter Bala, Mar 01 2014

Keywords

• nonn
• easy

Comments

Let P and Q be relatively prime integers. The Lucas sequence U(n) (which depends on P and Q) is an integer sequence that satisfies the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1. The sequence {U(n)}n>=1 is a strong divisibility sequence, i.e., gcd(U(n),U(m)) = |U(gcd(n,m))|. It follows that {U(n)} is a divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0.

It can be shown that if p and q are a pair of relatively prime positive integers, and if U(n) never vanishes, then the sequence {U(p*n)*U(q*n)/U(n)}n>=1 is a linear divisibility sequence of order 2*min(p,q). For a proof and a generalization of this result see the Bala link.

Here we take p = 3 and q = 4 with P = 1 and Q = -1, for which U(n) is the sequence of Fibonacci numbers, A000045, and normalize the sequence to have the initial term 1.

For other sequences of this type see A238601, A238602 and A238603. See also A238536.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• P. Bala, Divisibility sequences from strong divisibility sequences
• Wikipedia, Divisibility sequence
• Wikipedia, Fibonacci number
• Wikipedia, Lucas Sequence
• Index entries for linear recurrences with constant coefficients, signature (14,90,-350,90,14,-1).

Crossrefs

Cf. A000045, A215466, A238536, A238601, A238602, A238603.

Programs

Magma

[(Fibonacci(2*n) + (-1)^n*Fibonacci(4*n) + Fibonacci(6*n))/6: n in [1..30]]; // G. C. Greubel, Aug 07 2018

Maple

with(combinat):
seq(1/6*fibonacci(3*n)*fibonacci(4*n)/fibonacci(n), n = 1..20);

Mathematica

Table[(1/6)*(Fibonacci[2*n] + (-1)^n*Fibonacci[4*n] + Fibonacci[6*n]), {n, 1, 500}] (* G. C. Greubel, Aug 07 2018 *)
LinearRecurrence[{14,90,-350,90,14,-1},{1,28,408,7896,137555,2496144},20] (* Harvey P. Dale, Aug 26 2020 *)

PARI

vector(30, n, (fibonacci(2*n) + (-1)^n*fibonacci(4*n) + fibonacci(6*n))/6) \\ G. C. Greubel, Aug 07 2018

Formula

a(n) = (1/6)*Fibonacci(3*n)*Fibonacci(4*n)/Fibonacci(n).

a(n) = (1/6)*( Fibonacci(2*n) + (-1)^n*Fibonacci(4*n) + Fibonacci(6*n) ).

The sequence can be extended to negative indices when a(-n) = -a(n).

O.g.f. x*(1 + 14*x - 74*x^2 + 14*x^3 + x^4)/( (1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2) ).

Recurrence equation: a(n) = 14*a(n-1) + 90*a(n-2) - 350*a(n-3) + 90*a(n-4) + 14*a(n-5) - a(n-6).