A273622 a(n) = (1/3)*(Lucas(3*n) - Lucas(n)).

1, 5, 24, 105, 451, 1920, 8149, 34545, 146376, 620125, 2626999, 11128320, 47140601, 199691245, 845906424, 3583318305, 15179181851, 64300049280, 272379384749, 1153817597625, 4887649790376, 20704416783605, 87705316964399, 371525684705280

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

P. 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) = (A014448(n) - A000032(n))/3. - R. J. Mathar, Jun 07 2016

Maple

#A273622
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

Cf. A000032, A037451, A004146, A049682, A049683, A049684, A215465, A273623.

Sequence in context: A181305 A046724 A272578 * A271009 A270186 A008464

Adjacent sequences: A273619 A273620 A273621 * A273623 A273624 A273625

Keyword

nonn,easy

Author

Peter Bala, May 27 2016

Status

approved