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%I #20 Dec 18 2022 19:19:48
%S 1,1,2,7,8,26,32,85,128,292,512,1079,2048,4174,8192,16489,32768,65672,
%T 131072,262315,524288,1048786,2097152,4194557,8388608,16777516,
%U 33554432,67109215,134217728,268435862,536870912,1073742289,2147483648,4294967824,8589934592
%N Number of length n binary words that contain an even number of 0's or exactly two 1's.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-6,-3,6,1,-2).
%F G.f.: (1 - x - 3*x^2 + 6*x^3 - 3*x^4 - 2*x^5 - 3*x^6 + x^7)/( (1 - 2*x)*(1 - x^2)^3 ).
%F a(n) = (2^(1+n))/4 for n even; a(n) = (2^(1+n)-2*n+2*n^2)/4 for n odd. - _Colin Barker_, Jan 23 2014
%F E.g.f.: (1 + cosh(2*x) + x^2*sinh(x) + sinh(2*x))/2. - _Stefano Spezia_, Mar 20 2022
%e a(3)=7 because we have: 001, 010, 011, 100, 101, 110, 111.
%t nn=30;CoefficientList[Series[(1-x-3*x^2+6*x^3-3*x^4-2*x^5-3*x^6+x^7)/ ((1-2*x)*(1-x^2)^3),{x,0,nn}],x]
%t LinearRecurrence[{2,3,-6,-3,6,1,-2},{1,1,2,7,8,26,32,85},40] (* _Harvey P. Dale_, Dec 18 2022 *)
%Y Cf. A161680 (words containing exactly two 1's), A011782 (words containing an even number of 0's), A000384 (words containing an even number of 0's and exactly 2 1's).
%K nonn,easy
%O 0,3
%A _Geoffrey Critzer_, Jan 21 2014
%E More terms from _Colin Barker_, Jan 23 2014