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A108196 Expansion of (x-1)*(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 (list; graph; refs; listen; history; text; internal format)
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 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014
LINKS
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, 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(n-1) - 8*a(n-2) + 3*a(n-3) - a(n-4). - 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(n-1). Equivalently, a(n) = (-1)*U(n-1,1/2)*U(n-1,3/2), 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 4th-order linear divisibility sequences. (End)
MAPLE
seriestolist(series((x-1)*(x+1)/(8*x^2+1-3*x+x^4-3*x^3), x=0, 40));
MATHEMATICA
CoefficientList[Series[(x-1)(x+1)/(8x^2+1-3x+x^4-3x^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((x-1)*(x+1)/(8*x^2 +1 -3*x + x^4 - 3*x^3)) \\ G. C. Greubel, Aug 08 2017
CROSSREFS
Sequence in context: A215678 A186747 A083289 * A370015 A328341 A013460
KEYWORD
easy,sign
AUTHOR
Creighton Dement, Jul 23 2005
STATUS
approved

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Last modified May 30 18:38 EDT 2024. Contains 372971 sequences. (Running on oeis4.)