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%I #30 Mar 11 2024 03:48:45
%S -1,1,-1,3,-6,10,-15,23,-37,61,-100,162,-261,421,-681,1103,-1786,2890,
%T -4675,7563,-12237,19801,-32040,51842,-83881,135721,-219601,355323,
%U -574926,930250,-1505175,2435423,-3940597,6376021,-10316620,16692642,-27009261,43701901,-70711161,114413063
%N Expansion of (1-x)*(2*x^2+2*x+1) / ((x^2-x-1)*(x^2+x+1)).
%C Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[(.5'j + .5'k + .5j' + .5k' + .5'ij' - .5'ik' + .5'ji' + .5'ki')*(.5'i + .5'j + .5'k + .5e)]
%D Creighton Dement, Floretion Integer Sequences (work in progress).
%H Colin Barker, <a href="/A111734/b111734.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-1,0,1).
%F a(n) = ((-1)^n/2)*A007039(n); a(n) + a(n+1) + a(n+2) = ((-1)^n)*A000204(n).
%F a(n) = (-1)^n*(n*sum((binomial(n-2*m,2*m))/(n-2*m),m,0,floor((n-1)/2))). - _Vladimir Kruchinin_, Mar 10 2013
%F a(n) = -2*a(n-1) - a(n-2) + a(n-4) for n>4. - _Colin Barker_, May 18 2019
%F E.g.f.: exp(-x/2)*(cos(sqrt(3)*x/2) + cosh(sqrt(5)*x/2) + sqrt(3)*sin(sqrt(3)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/2. - _Stefano Spezia_, Aug 03 2022
%t CoefficientList[Series[(1-x)(2x^2+2x+1)/((x^2-x-1)(x^2+x+1)),{x,0,60}],x] (* or *) LinearRecurrence[{-2,-1,0,1},{-1,1,-1,3},60] (* _Harvey P. Dale_, Sep 01 2021 *)
%o (PARI) Vec(-x*(1 - x)*(1 + 2*x + 2*x^2) / ((1 + x - x^2)*(1 + x + x^2)) + O(x^40)) \\ _Colin Barker_, May 18 2019
%Y Cf. A000204, A007039.
%K easy,sign
%O 1,4
%A _Creighton Dement_, Nov 18 2005
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