A108196 Expansion of (x-1)*(x+1) / (8*x^2 + 1 - 3*x + x^4 - 3*x^3).

-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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Peter Bala, Linear divisibility sequences and Chebyshev polynomials

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.

Index entries for linear recurrences with constant coefficients, signature (3,-8,3,-1).

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] (* Harvey P. Dale, Dec 25 2012 *)
(* Alternative: *)
LinearRecurrence[{3, -8, 3, -1}, {-1, -3, 0, 21}, 40] (* Harvey P. Dale, Dec 25 2012 *)

Prog

(SageMath) [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

Cf. A000045, A100047.

Sequence in context: A215678 A186747 A083289 * A370015 A328341 A384896

Adjacent sequences: A108193 A108194 A108195 * A108197 A108198 A108199

Keyword

easy,sign,changed

Author

Creighton Dement, Jul 23 2005

Status

approved