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

Author

Creighton Dement, Jul 23 2005

Keywords

• easy
• sign

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
• 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 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

Crossrefs

Cf. A000045, A100047.

Programs

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 *)

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

Sage

[lucas_number1(n,3,1)*lucas_number1(n,1,1)*(-1) for n in range(1,33)] # Zerinvary Lajos, Jul 06 2008

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)