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%I #31 Mar 06 2024 15:47:24
%S 1,-2,5,-11,26,-53,104,-198,375,-700,1299,-2401,4432,-8167,15038,
%T -27676,50925,-93686,172337,-316999,583078,-1072473,1972612,-3628226,
%U 6673379,-12274288,22575967,-41523709,76374044,-140473803,258371642
%N Expansion of (1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4)/((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)).
%C A floretion-generated sequence involving tribonacci numbers. Formula for the g.f. provided by Alec Mihailovs. See sequence A104187 for the sequence generated without using a cyclic transformation (i->j, j->k, k->i), i.e. 1lesforrokseq (refer to FAMP Code).
%C Floretion Algebra Multiplication Program, FAMP Code: 1lesforcycrokseq[A*B} with A = - .5'ii' + .5'jj' + .5'kk' + .5e and B = + 'kj'. 1vesforcycrokseq[A*B] = A000004. ForType: 1A.
%H Colin Barker, <a href="/A104237/b104237.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,0,0,2,0,0,1).
%F G.f.: (1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4) / ((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)).
%F a(n) = -2*a(n-1) + 2*a(n-5) + a(n-8) for n>7. - _Colin Barker_, May 21 2019
%F a(n) = (1/4)*(2*(-1)^n*(A000073(n+5) + A000073(n+4)) - 2*A056594(n-1) - 3*(-1)^n*(2*n+3) - 1). - _G. C. Greubel_, Jul 08 2022
%t LinearRecurrence[{-2,0,0,0,2,0,0,1},{1,-2,5,-11,26,-53,104,-198},40] (* _Harvey P. Dale_, May 07 2016 *)
%o (PARI) Vec((1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4) / ((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)) + O(x^40)) \\ _Colin Barker_, May 21 2019
%o (SageMath)
%o @CachedFunction
%o def A000073(n):
%o if (n<3): return (n//2)
%o else: return A000073(n-1) + A000073(n-2) + A000073(n-3)
%o def A104237(n): return (1/4)*(2*(-1)^n*(A000073(n+5) + A000073(n+4)) - 2*i^(n-1)*(n%2) - 3*(-1)^n*(2*n+3) + 1)
%o [A104237(n) for n in (0..60)] # _G. C. Greubel_, Jul 08 2022
%Y Cf. A000073, A056594, A104187.
%K sign,easy
%O 0,2
%A _Creighton Dement_, Apr 02 2005
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