

A108196


Expansion of (x1)*(x+1) / (8*x^2 + 1  3*x + x^4  3*x^3).


4



1, 3, 0, 21, 55, 0, 377, 987, 0, 6765, 17711, 0, 121393, 317811, 0, 2178309, 5702887, 0, 39088169, 102334155, 0, 701408733, 1836311903, 0, 12586269025, 32951280099, 0, 225851433717, 591286729879, 0, 4052739537881
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OFFSET

0,2


COMMENTS

Terms (or their respective absolute values) appear to be contained in A000045.
Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 3, P2 = 6, Q = 1 of the 3 parameter family of 4thorder linear divisibility sequences found by Williams and Guy.  Peter Bala, Mar 25 2014


LINKS



FORMULA

a(0)=1, a(1)=3, a(2)=0, a(3)=21, a(n) = 3*a(n1)  8*a(n2) + 3*a(n3)  a(n4).  Harvey P. Dale, Dec 25 2012
The following formulas assume an offset of 1.
a(n) = (1)*A001906(n)*A010892(n1). Equivalently, a(n) = (1)*U(n1,1/2)*U(n1,3/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = (1)*bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 3/2; 1, 3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
The ordinary generating function is the Hadamard product of x/(1  x + x^2) and x/(1  3*x + x^2).
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4thorder linear divisibility sequences. (End)


MAPLE

seriestolist(series((x1)*(x+1)/(8*x^2+13*x+x^43*x^3), x=0, 40));


MATHEMATICA

CoefficientList[Series[(x1)(x+1)/(8x^2+13x+x^43x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 8, 3, 1}, {1, 3, 0, 21}, 40] (* Harvey P. Dale, Dec 25 2012 *)


PROG

(Sage) [lucas_number1(n, 3, 1)*lucas_number1(n, 1, 1)*(1) for n in range(1, 33)] # Zerinvary Lajos, Jul 06 2008
(PARI) x='x+O('x^50); Vec((x1)*(x+1)/(8*x^2 +1 3*x + x^4  3*x^3)) \\ G. C. Greubel, Aug 08 2017


CROSSREFS



KEYWORD

easy,sign


AUTHOR



STATUS

approved



