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%I #22 Sep 06 2023 16:19:06
%S 1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,1,10,1,11,1,12,1,13,1,14,1,15,1,16,
%T 1,17,1,18,1,19,1,20,1,21,1,22,1,23,1,24,1,25,1,26,1,27,1,28,1,29,1,
%U 30,1,31,1,32,1,33,1,34,1,35,1,36,1,37,1,38,1,39,1,40,1,41,1,42,1,43,1,44,1
%N Expansion of g.f. (1+x-x^3)/(1-x^2)^2.
%C Binomial transform is A111297. Binomial transform of [1,1,1,2,1,3,1,...] is A109975.
%C Essentially the same as A152271 and A133622. - _R. J. Mathar_, Mar 20 2009
%C Also defined by: a(0)=1; thereafter, a(n) = number of copies of a(n-1) in the list [a(0), a(1), ..., a(n-1)]. - _N. J. A. Sloane_, Feb 12 2016
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(k+1,n-k).
%F G.f.: Q(0)/x - 1/x, where Q(k)= 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 23 2013
%F E.g.f.: cosh(x) + (2 + x)*sinh(x)/2. - _Stefano Spezia_, Sep 06 2023
%t CoefficientList[Series[(1+x-x^3)/(1-x^2)^2,{x,0,100}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,1,2,1},100] (* _Harvey P. Dale_, Aug 17 2016 *)
%o (PARI) a(n)=1+!(n%2)*n/2 \\ _Jaume Oliver Lafont_, Mar 21 2009
%Y Cf. A109975, A111297, A133622, A152271.
%Y Related to A268696.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Mar 18 2009
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