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%I #18 Jul 10 2017 02:21:41
%S 1,2,5,10,13,26,29,58,61,122,125,250,253,506,509,1018,1021,2042,2045,
%T 4090,4093,8186,8189,16378,16381,32762,32765,65530,65533,131066,
%U 131069,262138,262141,524282,524285,1048570,1048573,2097146,2097149,4194298,4194301
%N Expansion of (1+2x)*(1+2*x^2)/((1-x)*(1+x)*(1-2*x^2)).
%H G. C. Greubel, <a href="/A185647/b185647.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-2).
%F a(n) = a(n-1)*2 if n odd.
%F a(n) = a(n-1)+3 if n even.
%F a(2n) = 2^(n+2)-3 = A036563(n+2).
%F a(2n+1) = 2^(n+3)-6 = A131130(n+1).
%F a(n) = 3*a(n-2) - 2*a(n-4) with a(0)=1, a(1)=2, a(2)=5, a(3)=10.
%t LinearRecurrence[{0, 3, 0, -2}, {1, 2, 5, 10}, 50] (* _G. C. Greubel_, Jul 09 2017 *)
%o (PARI) x='x+O('x^50); Vec((1+2*x)*(1+2*x^2)/((1-x)*(1+x)*(1-2*x^2))) \\ _G. C. Greubel_, Jul 09 2017
%Y Cf. A036563, A131130
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Apr 23 2013