A107363 - OEIS

%I #16 Jun 10 2024 12:47:22

%S 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,

%T 233,-521,233,610,0,1597,987,-2207,987,2584,0,6765,4181,-9349,4181,

%U 10946,0,28657,17711,-39603,17711,46368,0,121393,75025,-167761,75025,196418,0,514229,317811,-710647,317811,832040,0

%N 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)).

%C 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).

%C 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')]

%H Colin Barker, <a href="/A107363/b107363.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,4,0,0,0,0,0,1).

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

%F From _Colin Barker_, May 11 2019: (Start)

%F 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)).

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

%t 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 *)

%o (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

%Y Cf. A000045, A048876.

%K sign,easy

%O 0,5

%A _Creighton Dement_, May 24 2005