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A238600 A sixth-order linear divisibility sequence related to the Fibonacci numbers: a(n) := (1/6)*Fibonacci(3*n)*Fibonacci(4*n)/Fibonacci(n). 6
1, 28, 408, 7896, 137555, 2496144, 44599477, 801617712, 14375440584, 258018516140, 4629531440711, 83076469908768, 1490726895438793, 26750144944686436, 480010941060482040, 8613453244178393184, 154562103244937408987, 2773504708179098411952 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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).

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 *)

PROG

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

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

CROSSREFS

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

Sequence in context: A022592 A323973 A121798 * A278009 A233333 A271793

Adjacent sequences:  A238597 A238598 A238599 * A238601 A238602 A238603

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Mar 01 2014

STATUS

approved

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Last modified June 15 21:52 EDT 2021. Contains 345053 sequences. (Running on oeis4.)