A215466 Expansion of x*(1-4*x+x^2) / ( (x^2-7*x+1)*(x^2-3*x+1) ).

0, 1, 6, 38, 252, 1705, 11628, 79547, 544824, 3733234, 25585230, 175356611, 1201893336, 8237850373, 56462937882, 387002396990, 2652553009008, 18180866487757, 124613506702404, 854113665498719, 5854182112700460

Offset: 0

Author

R. J. Mathar, Aug 11 2012

Keywords

• nonn
• easy

Comments

From Peter Bala, Aug 05 2019: (Start)

Let U(n;P,Q), where P and Q are integer parameters, denote the Lucas sequence of the first kind. Then, excluding the case P = -1, the sequence ( U(n;P,1) + U(2*n;P,1) )/(P + 1) is a fourth-order linear divisibility sequence with o.g.f. x*(1 - 2*(P - 1)*x + x^2)/((1 - P*x + x^2)*(1 - (P^2 - 2)*x + x^2)). This is the case P = 3. See A000027 (P = 2), A165998 (P = -2) and A238536 (P = -3).

More generally, the sequence U(n;P,1) + U(2*n;P,1) + ... + U(k*n;P,1) is a linear divisibility sequence of order 2*k. As an example, see A273625 (P = 3, k = 3 and then sequence normalized with initial term 1). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Wikipedia, Lucas sequence
• Peter Bala, Divisibility sequences from strong divisibility sequences
• 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.
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Odd and even linear divisibility sequences of order 4, INTEGERS, 2015, #A33.
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (10,-23,10,-1).

Crossrefs

Cf. A000032, A000045, A001906, A033888, A165998, A215465, A238536, A273625.

Programs

Magma

I:=[0,1,6,38]; [n le 4 select I[n] else 10*Self(n-1)-23*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 23 2012

Magma

/* By definition: */ m:=20; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1-4*x+x^2)/((x^2-7*x+1)*(x^2-3*x+1)))); // Bruno Berselli, Dec 24 2012

Maple

A215466 := proc(n)
    if type(n,'even') then
        A000032(n)*combinat[fibonacci](3*n)/4 ;
    else
        combinat[fibonacci](n)*A000032(3*n)/4 ;
    end if;
end proc:

Mathematica

CoefficientList[Series[x*(1 - 4*x + x^2)/((x^2 - 7*x + 1)*(x^2 - 3*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 23 2012 *)
LinearRecurrence[{10,-23,10,-1},{0,1,6,38},30] (* Harvey P. Dale, Nov 02 2015 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,10,-23,10]^n*[0;1;6;38])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

PARI

{a(n) = my(w=quadgen(5)^(2*n)); imag(w^2+w)/4}; /* Michael Somos, Dec 29 2022 */

Formula

a(n) = L(n)*F(3n)/4 if n even, = F(n)*L(3n)/4 if n odd, where L=A000032, F=A000045.

a(n) = 3*A004187(n)/4 + A001906(n)/4.

a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4), a(0)=0, a(1)=1, a(2)=6, a(3)=38. - Harvey P. Dale, Nov 02 2015

a(n) = (1/4)*(Fibonacci(2*n) + Fibonacci(4*n)) = (1/4)*(A001906(n) + A033888(n)). - Peter Bala, Aug 05 2019

E.g.f.: exp(5*x/2)*(cosh(x)+exp(x)*cosh(sqrt(5)*x))*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Aug 17 2019

a(n) = -a(-n) for all n in Z. - Michael Somos, Dec 29 2022