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A273622 a(n) = (1/3)*(Lucas(3*n) - Lucas(n)). 9
1, 5, 24, 105, 451, 1920, 8149, 34545, 146376, 620125, 2626999, 11128320, 47140601, 199691245, 845906424, 3583318305, 15179181851, 64300049280, 272379384749, 1153817597625, 4887649790376, 20704416783605, 87705316964399, 371525684705280 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified October 22 19:53 EDT 2019. Contains 328319 sequences. (Running on oeis4.)