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%I #18 Sep 08 2022 08:45:23
%S 1,5,11,30,71,175,421,1020,2461,5945,14351,34650,83651,201955,487561,
%T 1177080,2841721,6860525,16562771,39986070,96534911,233055895,
%U 562646701,1358349300,3279345301,7917039905,19113425111,46143890130,111401205371,268946300875
%N Expansion of (1 +3*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
%C Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e
%H Colin Barker, <a href="/A114688/b114688.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F G.f.: (1 +3*x -x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
%F a(0)=1, a(1)=5, a(2)=11, a(3)=30, a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4). - _Harvey P. Dale_, Dec 18 2012
%F a(n) = (-6 - 6*(-1)^n + 5*sqrt(2)*( (1+sqrt(2))^(1+n) - (1-sqrt(2))^(1+n) ))/8. - _Colin Barker_, May 26 2016
%F a(n) = (10*A000129(n+1) - 3*(1 + (-1)^n))/4. - _G. C. Greubel_, May 24 2021
%p Pell:= proc(n) option remember;
%p if n<2 then n
%p else 2*Pell(n-1) + Pell(n-2)
%p fi; end:
%p seq((10*Pell(n+1) -3*(1+(-1)^n))/4, n=0..40); # _G. C. Greubel_, May 24 2021
%t CoefficientList[Series[(-1-3x+x^2)/((1-x)(x+1)(x^2+2x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-2,-1},{1,5,11,30},40] (* _Harvey P. Dale_, Dec 18 2012 *)
%o (PARI) Vec((-1-3*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^50)) \\ _Colin Barker_, May 26 2016
%o (Magma) I:=[1,5,11,30]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // _G. C. Greubel_, May 24 2021
%o (Sage) [(10*lucas_number1(n+1,2,-1) -3*(1+(-1)^n))/4 for n in (0..30)] # _G. C. Greubel_, May 24 2021
%Y Cf. A000129, A100828, A114647, A114689, A114695, A114696, A114697.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Feb 18 2006
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