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%I #34 Feb 21 2021 04:09:29
%S 1,0,4,12,48,192,768,3072,12288,49152,196608,786432,3145728,12582912,
%T 50331648,201326592,805306368,3221225472,12884901888,51539607552,
%U 206158430208,824633720832,3298534883328,13194139533312,52776558133248
%N Expansion of (1 - 4*x + 4*x^2 - 4*x^3)/(1 - 4*x).
%C Partial sums are A092896.
%H G. C. Greubel, <a href="/A092898/b092898.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F a(n+2) = 4 * A002001(n).
%F a(n) = (3*4^n + 13*0^n)/16 + Sum_{k=0..n} binomial(n, k)*(-1)^k*(3*k/4 + k*(k-1)/2).
%F G.f.: 1 - x + 8*x^2 + 2*x/G(0), where G(k) = 1 + 1/(1 - x*(3*k+4)/(x*(3*k+7) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 11 2013
%F a(n) = A110594(n-1) for n >= 2. - _Georg Fischer_, Nov 03 2018
%F From _G. C. Greubel_, Feb 21 2021: (Start)
%F a(n) = (3*4^n +16*[n=2] -12*[n=1] +13*0^n)/16.
%F E.g.f.: (13 -12*x + 8*x^2 + 3*exp(4*x))/16. (End)
%p a:= n-> 3*4^n/16+13*0^n/16+add(binomial(n,k)*(-1)^k*(3*k/4+k*(k-1)/2), k=0..n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 03 2018
%t Join[{1, 0, 4}, LinearRecurrence[{4}, {12}, 22]] (* _Jean-François Alcover_, Sep 16 2019 *)
%o (PARI) Vec((1 -4*x +4*x^2 -4*x^3)/(1-4*x) + O(x^30)) \\ _Andrew Howroyd_, Nov 03 2018
%o (Sage) [1,0,4]+[3*4^(n-2) for n in (3..30)] # _G. C. Greubel_, Feb 21 2021
%o (Magma) [1,0,4] cat [3*4^(n-2): n in [3..30]]; // _G. C. Greubel_, Feb 21 2021
%Y Cf. A002001, A092896, A110594.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Mar 12 2004
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