OFFSET
1,2
COMMENTS
This is a divisibility sequence, that is, a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4. Cf. A273623.
More generally, for distinct integers r and s with r == s (mod 2), the sequence Lucas(r*n) - Lucas(s*n) is a fourth-order divisibility sequence. When r is even (resp. odd) the normalized sequence (Lucas(r*n) - Lucas(s*n))/(Lucas(r) - Lucas(s)), with initial term equal to 1, has the o.g.f. x*(1 - x^2)/( (1 - Lucas(r)*x + x^2)*(1 - Lucas(s)*x + x^2) ) (resp. x*(1 + x^2)/( (1 - Lucas(r)*x - x^2)*(1 - Lucas(s)*x - x^2) )) and belongs to the 3-parameter family of fourth-order divisibility sequences found by Williams and Guy, with parameter values P1 = (Lucas(r) + Lucas(s)), P2 = Lucas(r)*Lucas(s) and Q = 1 (resp. Q = -1). For particular cases see A004146 (r = 2, s = 0), A049684 (r = 4, s = 0), A215465 (r = 4, s = 2), A049683 (r = 6, s = 0), A049682 (r = 8, s = 0) and A037451 (r = 3, s = -1).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Bala, Lucas sequences and divisibility sequences
Spirit Karcher and Mariah Michael, Prime Factors and Divisibility of Sums of Powers of Fibonacci and Lucas Numbers, Amer. J. of Undergraduate Research (2021) Vol. 17, Issue 4, 59-69.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, 7 (5) (2011), 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (5,-2,-5,-1).
FORMULA
a(n) = (1/3)*( (2 + sqrt(5))^n + (2 - sqrt(5))^n - ((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n ).
a(n) = -a(-n).
a(n) = 5*a(n-1) - 2*a(n-2) - 5*a(n-3) - a(n-4).
O.g.f.: x*(1 + x^2)/((1 - x - x^2 )*(1 - 4*x - x^2)).
a(n) = Fibonacci(n) + Sum_{k=1..n} Fibonacci(n-k)*Lucas(3*k). - Yomna Bakr and Greg Dresden, Jun 16 2024
E.g.f.: (2*exp(2*x)*cosh(sqrt(5)*x) - 2*exp(x/2)*cosh(sqrt(5)*x/2))/3. - Stefano Spezia, Jun 17 2024
MAPLE
with(combinat):
Lucas := n->fibonacci(n+1) + fibonacci(n-1):
seq(1/3*(Lucas(3*n) - Lucas(n)), n = 1..24);
MATHEMATICA
LinearRecurrence[{5, -2, -5, -1}, {1, 5, 24, 105}, 100] (* G. C. Greubel, Jun 02 2016 *)
Table[1/3 (LucasL[3 n] - LucasL[n]), {n, 1, 30}] (* Vincenzo Librandi, Jun 02 2016 *)
PROG
(Magma) [1/3*(Lucas(3*n) - Lucas(n)): n in [1..25]]; // Vincenzo Librandi, Jun 02 2016
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, -5, -2, 5]^(n-1)*[1; 5; 24; 105])[1, 1] \\ Charles R Greathouse IV, Jun 07 2016
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Peter Bala, May 27 2016
STATUS
approved