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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).

%I A238536 #40 Sep 08 2022 08:46:07
%S A238536 1,12,68,504,3355,23256,158717,1089648,7463884,51170460,350695511,
%T A238536 2403786672,16475579353,112925875764,774003961940,5305106018016,
%U A238536 36361727272627,249227013404808,1708227291909269,11708364225400920,80250321774226396,550043889533755332,3770056901455017263
%N A238536 A fourth-order linear divisibility sequence related to the Fibonacci numbers: a(n) = (1/2)*Fibonacci(3*n)*Lucas(n).
%C A238536 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.
%D A238536 S. Koshkin, Non-classical linear divisibility sequences ..., Fib. Q., 57 (No. 1, 2019), 68-80.
%H A238536 G. C. Greubel, <a href="/A238536/b238536.txt">Table of n, a(n) for n = 1..1185</a>
%H A238536 Peter Bala, <a href="/A238536/a238536.pdf">A family of linear divisibility sequences of order four</a>
%H A238536 E. L. Roettger and H. C. Williams, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Roettger/roettger12.html">Appearance of Primes in Fourth-Order Odd Divisibility Sequences</a>, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
%H A238536 Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisibility_sequence">Divisibility sequence</a>
%H A238536 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>
%H A238536 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,19,4,-1).
%F A238536 a(n) = (1/2)*Fibonacci(3*n)*Lucas(n) = (1/2)*A000045(3*n)*A000032(n).
%F A238536 a(n) = (1/2)*Fibonacci(2*n)*Fibonacci(3*n)/Fibonacci(n).
%F A238536 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 ).
%F A238536 The sequence can be extended to negative indices by setting a(-n) = -a(n).
%F A238536 O.g.f. x*(1 + 8*x + x^2)/( (1 + 3*x + x^2)*(1 - 7*x + x^2) ).
%F A238536 Recurrence equation: a(n) = 4*a(n-1) + 19*a(n-2) + 4*a(n-3) - a(n-4).
%F A238536 a(n) = (1/2) * (Fibonacci(4*n) + (-1)^n*Fibonacci(2*n)). - _Ralf Stephan_, Mar 01 2014
%p A238536 with(combinat): lucas:= n->fibonacci(n+1)+ fibonacci(n-1):
%p A238536 seq(1/2*lucas(n)*fibonacci(3*n), n = 1..24);
%t A238536 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 *)
%o A238536 (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
%o A238536 (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
%Y A238536 Cf. A000032, A000045, A127595, A215466, A238537, A238538.
%K A238536 nonn,easy
%O A238536 1,2
%A A238536 _Peter Bala_, Feb 28 2014