A107363 Expansion of (1 - x)*(1 + x)^2*(1 + x^2)*(1 - x^2 + 2*x^3 + x^4) / ((1 - x^2 - x^4)*(1 + x^2 + 2*x^4 - x^6 + x^8)).

1, 1, -1, 1, 2, 0, 5, 3, -7, 3, 8, 0, 21, 13, -29, 13, 34, 0, 89, 55, -123, 55, 144, 0, 377, 233, -521, 233, 610, 0, 1597, 987, -2207, 987, 2584, 0, 6765, 4181, -9349, 4181, 10946, 0, 28657, 17711, -39603, 17711, 46368, 0, 121393, 75025, -167761, 75025, 196418, 0, 514229, 317811, -710647, 317811, 832040, 0

Offset: 0

Author

Creighton Dement, May 24 2005

Keywords

• sign
• easy

Comments

Conjectures: { Fib(n) | n in naturals } = { a(n) | n in naturals, a(n) >= 0 } = { a(n) | n in naturals, n not of the form 6*n+2 } (naturals include 0).

Floretion Algebra Multiplication Program, FAMP Code: 4teszapseq[(- .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*( + .5'j + .5i' + .5'ik' + .5'jk' + .5'ki' + .5'kj')]

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,4,0,0,0,0,0,1).

Crossrefs

Cf. A000045, A048876.

Programs

Mathematica

CoefficientList[Series[(1-x)(1+x)^2(1+x^2)(1-x^2+2x^3+x^4)/((1-x^2-x^4)(1+x^2+2x^4-x^6+x^8)),{x,0,80}],x] (* or *) LinearRecurrence[{0,0,0,0,0,4,0,0,0,0,0,1},{1,1,-1,1,2,0,5,3,-7,3,8,0},80] (* Harvey P. Dale, Jun 10 2024 *)

PARI

Vec((1 - x)*(1 + x)^2*(1 + x^2)*(1 - x^2 + 2*x^3 + x^4) / ((1 - x^2 - x^4)*(1 + x^2 + 2*x^4 - x^6 + x^8)) + O(x^55)) \\ Colin Barker, May 11 2019

Formula

a(6*n+2) = - A048876(n) (Generalized Pellian with second term of 7), conjecture.

From Colin Barker, May 11 2019: (Start)

G.f.: (1 - x)*(1 + x)^2*(1 + x^2)*(1 - x^2 + 2*x^3 + x^4) / ((1 - x^2 - x^4)*(1 + x^2 + 2*x^4 - x^6 + x^8)).

a(n) = 4*a(n-6) + a(n-12) for n>11. (End)