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, 1573808055889201, 6666757908429845

Offset: 1

Author

Peter Bala, May 27 2016

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[1/3*(Lucas(3*n) - Lucas(n)): n in [1..25]]; // Vincenzo Librandi, Jun 02 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 *)

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

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

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