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%I #25 Mar 09 2024 14:54:36
%S 2,5,14,36,95,248,650,1701,4454,11660,30527,79920,209234,547781,
%T 1434110,3754548,9829535,25734056,67372634,176383845,461778902,
%U 1208952860,3165079679,8286286176,21693778850,56795050373,148691372270,389279066436,1019145827039
%N Expansion of (x+2) / ((x+1)*(x^2-3*x+1)).
%C A floretion-generated sequence relating Fibonacci numbers.
%C Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 2dia[I]forseq[ + .5'i + .5'ii' + .5'ij' + .5'ik' ], 2dia[J]forseq = 2dia[K]forseq = A001654, mixforseq = A001519, tesforseq = A099016, vesforseq = A000004. Identity used: dia[I] + dia[J] + dia[K] + mix + tes = ves
%H Colin Barker, <a href="/A102714/b102714.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), a(0) = 2, a(1) = 5, a(2) = 14.
%F a(n) + a(n+1) = A100545(n).
%F a(n) + 2*a(n+1) + a(n+2) = A055849(n+2).
%F a(n) + 2*A001654(n) - A099016(n+2) + 2*A001519(n) = 0.
%F a(n) = (2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5. - _Colin Barker_, Oct 01 2016
%F a(n) = (-1)^n +9*A001906(n+1) -A001906(n) . - _R. J. Mathar_, Sep 11 2019
%t CoefficientList[Series[(x+2)/((x+1)(x^2-3x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-1},{2,5,14},30] (* _Harvey P. Dale_, Apr 22 2012 *)
%o (PARI) a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5) \\ _Colin Barker_, Oct 01 2016
%o (PARI) Vec((x+2)/((x+1)*(x^2-3*x+1)) + O(x^40)) \\ _Colin Barker_, Oct 01 2016
%Y Cf. A001519, A099016, A001654, A100545, A055849, A000004.
%K easy,nonn
%O 0,1
%A _Creighton Dement_, Feb 06 2005
%E Corrected by _T. D. Noe_, Nov 02 2006, Nov 07 2006
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