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A238536
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A fourth-order linear divisibility sequence related to the Fibonacci numbers: a(n) = (1/2)*Fibonacci(3*n)*Lucas(n).
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11
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1, 12, 68, 504, 3355, 23256, 158717, 1089648, 7463884, 51170460, 350695511, 2403786672, 16475579353, 112925875764, 774003961940, 5305106018016, 36361727272627, 249227013404808, 1708227291909269, 11708364225400920, 80250321774226396, 550043889533755332, 3770056901455017263
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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S. Koshkin, Non-classical linear divisibility sequences ..., Fib. Q., 57 (No. 1, 2019), 68-80.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..1185
Peter Bala, A family of linear divisibility sequences of order four
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
Wikipedia, Divisibility sequence
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (4,19,4,-1).
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FORMULA
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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
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MAPLE
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with(combinat): lucas:= n->fibonacci(n+1)+ fibonacci(n-1):
seq(1/2*lucas(n)*fibonacci(3*n), n = 1..24);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000032, A000045, A127595, A215466, A238537, A238538.
Sequence in context: A059585 A213547 A050484 * A096425 A212753 A210427
Adjacent sequences: A238533 A238534 A238535 * A238537 A238538 A238539
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KEYWORD
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nonn,easy
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AUTHOR
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Peter Bala, Feb 28 2014
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STATUS
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approved
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