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A174618 For n odd a(n) = a(n-2) + a(n-3), for n even a(n) = a(n-2) + a(n-5); with a(1) = 0, a(2) = 1.

%I A174618 #24 Oct 25 2024 09:33:34
%S A174618 0,1,0,1,1,1,2,1,3,2,4,4,6,7,10,11,17,17,28,27,45,44,72,72,116,117,
%T A174618 188,189,305,305,494,493,799,798,1292,1292,2090,2091,3382,3383,5473,
%U A174618 5473,8856,8855,14329,14328,23184,23184,37512,37513,60696
%N A174618 For n odd a(n) = a(n-2) + a(n-3), for n even a(n) = a(n-2) + a(n-5); with a(1) = 0, a(2) = 1.
%C A174618 Combination a(2n)=A005252(n-1) and a(2n+1)=A024490(n). Consecutive pairs add up to A000045 and subtract to A010892. If a(1)= 1 formula gives: A103609.
%H A174618 Harvey P. Dale, <a href="/A174618/b174618.txt">Table of n, a(n) for n = 1..1000</a>
%H A174618 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1,0,0,0,1).
%F A174618 G.f.: x^2*(1-x^2+x^3) / ( (1-x^2+x^4)*(1-x^2-x^4) ). - _R. J. Mathar_, Jan 27 2011
%F A174618 a(n) = (1/2)*(A110161(n-3) - A110161(n-2) + A079977(n-2) + A079977(n-1)). - _G. C. Greubel_, Oct 23 2024
%e A174618 As consecutive pairs: (0,1),(0,1),(1,1),(2,1),(3,2),(4,4),...
%t A174618 nxt[{n_,a_,b_,c_,d_,e_}]:={n+1,b,c,d,e,If[EvenQ[n],d+c,d+a]}; NestList[nxt,{5,0,1,0,1,1},50][[All,2]] (* or *) LinearRecurrence[ {0,2,0,-1,0,0,0,1},{0,1,0,1,1,1,2,1},60] (* _Harvey P. Dale_, Nov 15 2019 *)
%o A174618 (Magma)
%o A174618 R<x>:=PowerSeriesRing(Integers(), 70);
%o A174618 [0] cat Coefficients(R!( x^2*(1-x^2+x^3)/((1-x^2+x^4)*(1-x^2-x^4)) )); // _G. C. Greubel_, Oct 23 2024
%o A174618 (SageMath)
%o A174618 def A174618(n): return (kronecker(12,n-3) - kronecker(12,n-2) + ((n+1)%2)*fibonacci(n//2) + (n%2)*fibonacci((n+1)//2))//2
%o A174618 [A174618(n) for n in range(1,71)] # _G. C. Greubel_, Oct 23 2024
%Y A174618 Cf. A000045, A005252, A010892, A024490, A079977, A110161.
%K A174618 nonn,easy
%O A174618 1,7
%A A174618 _Mark Dols_, Mar 23 2010