This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A178945 #22 Feb 20 2025 12:53:18
%S A178945 1,2,7,16,42,96,228,512,1160,2560,5648,12288,26656,57344,122944,
%T A178945 262144,557184,1179648,2490624,5242880,11010560,23068672,48235520,
%U A178945 100663296,209717248,436207616,905973760,1879048192,3892322304,8053063680,16643014656
%N A178945 Expansion of x*(1-x)^2/( (1-2*x^2)*(1-2*x)^2).
%H A178945 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-8,8).
%F A178945 a(2n+1) = ( A001787(2n+1)+A077957(2n))/2.
%F A178945 a(2n) = A001787(2n)/2.
%F A178945 a(n) = 2^(n-2)*n + 2^(n/2-5/2)*(1-(-1)^n).
%F A178945 a(n) = +4*a(n-1) -2*a(n-2) -8*a(n-3) +8*a(n-4).
%F A178945 G.f.: x*(S(x)^2 + S(x^2))/2 where S(x) is the g.f. for A000079.
%e A178945 (1, 4, 12, 32, 80, 192, 448, 1024,...) +
%e A178945 ..(1, 0,..2,..0,..4,...0,...8,....0...) =
%e A178945 ..(2, 4, 14, 32, 84, 192, 456, 1024,...). Then dividing the sum by 2 we obtain:
%e A178945 ..(1, 2, 7, 16, 42, 96, 228,...).
%t A178945 CoefficientList[Series[x (1-x)^2/((1-2x^2)(1-2x)^2),{x,0,50}],x] (* or *) LinearRecurrence[{4,-2,-8,8},{0,1,2,7},50] (* _Harvey P. Dale_, Dec 29 2023 *)
%Y A178945 Cf. A000079, A001787, A077957, column k=2 of A290222.
%K A178945 nonn,easy
%O A178945 1,2
%A A178945 _Gary W. Adamson_, Dec 30 2010