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A238536 A fourth-order linear divisibility sequence related to the Fibonacci numbers: a(n) = (1/2)*Fibonacci(3*n)*Lucas(n). 11
1, 12, 68, 504, 3355, 23256, 158717, 1089648, 7463884, 51170460, 350695511, 2403786672, 16475579353, 112925875764, 774003961940, 5305106018016, 36361727272627, 249227013404808, 1708227291909269, 11708364225400920, 80250321774226396, 550043889533755332, 3770056901455017263 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let P and Q be integers. The Lucas sequences U(n) and V(n) (which depend on P and Q) are a pair of integer sequences that satisfy the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1 and V(0) = 2, V(1) = P, respectively. The sequence {U(n)} n >= 1 is a linear divisibility sequence of order 2, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0. In general, V(n) is not a divisibility sequence. However, it can be shown that if p >= 3 is an odd integer then the sequence {U(p*n)*V(n)} n >= 1 is a linear divisibility sequence of order 4. For a proof and a generalization of this result see the Bala link. Here we take p = 3 with P = 1 and Q = -1, for which U(n) is the sequence of Fibonacci numbers, A000045, V(n) is the sequence of Lucas numbers, A000032, and normalize the sequence to have the initial term 1. For other sequences of this type see A238537 and A238538.

REFERENCES

S. Koshkin, Non-classical linear divisibility sequences ..., Fib. Q., 57 (No. 1, 2019), 68-80.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1185

Peter Bala, A family of linear divisibility sequences of order four

Wikipedia, Divisibility sequence

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (4,19,4,-1).

FORMULA

a(n) = (1/2)*Fibonacci(3*n)*Lucas(n) = (1/2)*A000045(3*n)*A000032(n).

a(n) = (1/2)*Fibonacci(2*n)*Fibonacci(3*n)/Fibonacci(n).

a(n) = (1/(2*sqrt(5)))*( ((7 + 3*sqrt(5))/2)^n - ((7 - 3*sqrt(5))/2)^n + (-1)^n*((3 + sqrt(5))/2)^n - (-1)^n*((3 - sqrt(5))/2)^n ).

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

O.g.f. x*(1 + 8*x + x^2)/( (1 + 3*x + x^2)*(1 - 7*x + x^2) ).

Recurrence equation: a(n) = 4*a(n-1) + 19*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = (1/2) * (Fibonacci(4*n) + (-1)^n*Fibonacci(2*n)). - Ralf Stephan, Mar 01 2014

MAPLE

with(combinat): lucas:= n->fibonacci(n+1)+ fibonacci(n-1):

seq(1/2*lucas(n)*fibonacci(3*n), n = 1..24);

MATHEMATICA

Table[Fibonacci(3*n)*Lucas(n)/2, {n, 1, 30}] (* or *) Join[{1}, LinearRecurrence[{4, 19, 4, -1}, {12, 68, 504, 3355}, 30]] (* G. C. Greubel, Dec 25 2017 *)

PROG

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 4, 19, 4]^(n-1)*[1; 12; 68; 504])[1, 1] \\ Charles R Greathouse IV, Oct 07 2016

(MAGMA) I:=[12, 68, 504, 3355]; [1] cat [n le 4 select I[n] else 4*Self(n-1) + 19*Self(n-2) + 4*Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 25 2017

CROSSREFS

Cf. A000032, A000045, A127595, A215466, A238537, A238538.

Sequence in context: A059585 A213547 A050484 * A096425 A212753 A210427

Adjacent sequences:  A238533 A238534 A238535 * A238537 A238538 A238539

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Feb 28 2014

STATUS

approved

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Last modified June 25 04:01 EDT 2021. Contains 345450 sequences. (Running on oeis4.)