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Expansion of (1 + x + x^2)/(1 - x^2 - 2*x^3).
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%I #29 Mar 23 2023 06:16:09

%S 1,1,2,3,4,7,10,15,24,35,54,83,124,191,290,439,672,1019,1550,2363,

%T 3588,5463,8314,12639,19240,29267,44518,67747,103052,156783,238546,

%U 362887,552112,839979,1277886,1944203,2957844,4499975,6846250,10415663

%N Expansion of (1 + x + x^2)/(1 - x^2 - 2*x^3).

%C A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj')

%C Starting with offset 1 the sequence appears to be the INVERT transform of (1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, ...). - _Gary W. Adamson_, Aug 27 2016

%H G. C. Greubel, <a href="/A159288/b159288.txt">Table of n, a(n) for n = 0..1000</a>

%H Creighton Dement, <a href="http://fumba.eu/sitelayout/Floretion.php">Online Floretion Multiplier</a> [broken link]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2).

%F a(n) = A159287(n) + A159287(n+1) + A159287(n+2). - _R. J. Mathar_, Apr 10 2009

%F a(n) = a(n-2) + 2*a(n-3) for n>2. - _Colin Barker_, Apr 29 2019

%F a(n)= A052947(n) + A052947(n-1) +A052947(n-2). - _R. J. Mathar_, Mar 23 2023

%t CoefficientList[Series[(1+x+x^2)/(1-x^2-2x^3),{x,0,50}],x] (* _Harvey P. Dale_, Mar 09 2011 *)

%t LinearRecurrence[{0, 1, 2}, {1, 1, 2}, 50] (* _G. C. Greubel_, Jun 27 2018 *)

%o (PARI) a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;1;2])[1,1] \\ _Charles R Greathouse IV_, Aug 27 2016

%o (PARI) Vec((1 + x + x^2) / (1 - x^2 - 2*x^3) + O(x^40)) \\ _Colin Barker_, Apr 29 2019

%o (Magma) I:=[1, 1, 2]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jun 27 2018

%Y Cf. A159284, A159285, A159286, A159287.

%K easy,nonn

%O 0,3

%A _Creighton Dement_, Apr 08 2009