

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,8,3,1).
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some fourthorder linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 12551277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume


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
From Peter Bala, Mar 25 2014: (Start)
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 xrange(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

Cf. A000045, A100047.
Sequence in context: A215678 A186747 A083289 * A013460 A013388 A057379
Adjacent sequences: A108193 A108194 A108195 * A108197 A108198 A108199


KEYWORD

easy,sign


AUTHOR

Creighton Dement, Jul 23 2005


STATUS

approved



