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%I #20 Apr 01 2024 09:13:24
%S 0,0,0,0,8,20,24,8,0,0,0,0,128,320,384,128,0,0,0,0,2048,5120,6144,
%T 2048,0,0,0,0,32768,81920,98304,32768,0,0,0,0,524288,1310720,1572864,
%U 524288,0,0,0,0,8388608,20971520,25165824,8388608,0,0,0,0,134217728,335544320
%N Expansion of 4*x^4*(2 + x)/(1 - 2*x + 2*x^2 - 4*x^4 + 8*x^5 - 8*x^6).
%C a(n) = 0 iff n == {0, 1, 2 or 3} (mod 8). - _Robert G. Wilson v_, Nov 12 2004
%H G. C. Greubel, <a href="/A100212/b100212.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,0,4,-8,8).
%F a(8n+4) = a(8n+7) = 2^(4n+3), a(8n+5) = (5/2)*2^(4n+3), a(8n+6) = 3*2^(4n+3), a(8n+8) = 0, a(8n+9) = 0, a(8n+10) = 0, a(8n+11) = 0.
%F (a(n)) = negseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)
%F a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=8, a(5)=20, a(n) = 2*a(n-1) - 2*a(n-2) + 4*a(n-4) - 8*a(n-5) + 8*a(n-6). - _Harvey P. Dale_, Oct 10 2012
%t CoefficientList[ Series[4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6), {x, 0, 55}], x] (* _Robert G. Wilson v_, Nov 12 2004 *)
%t LinearRecurrence[{2,-2,0,4,-8,8},{0,0,0,0,8,20},60] (* _Harvey P. Dale_, Oct 10 2012 *)
%o (PARI) Vec(4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( 4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6) )); // _G. C. Greubel_, Apr 01 2024
%o (SageMath)
%o def A100212_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 4*x^4*(2+x)/(1-2*x+2*x^2-4*x^4+8*x^5-8*x^6) ).list()
%o A100212_list(60) # _G. C. Greubel_, Apr 01 2024
%Y Cf. A038503, A009116, A100213.
%K nonn,easy
%O 0,5
%A _Creighton Dement_, Nov 08 2004
%E More terms from _Robert G. Wilson v_, Nov 12 2004
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