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A117909 Count, inserting 0 after every even number.

%I A117909 #15 Sep 08 2022 08:45:24
%S A117909 1,2,0,3,4,0,5,6,0,7,8,0,9,10,0,11,12,0,13,14,0,15,16,0,17,18,0,19,20,
%T A117909 0,21,22,0,23,24,0,25,26,0,27,28,0,29,30,0,31,32,0,33,34,0,35,36,0,37,
%U A117909 38,0,39,40,0,41,42,0,43,44,0,45,46,0,47,48,0,49,50,0,51,52,0,53,54,0
%N A117909 Count, inserting 0 after every even number.
%C A117909 Row sums of A117908.
%H A117909 G. C. Greubel, <a href="/A117909/b117909.txt">Table of n, a(n) for n = 0..1000</a>
%H A117909 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1)
%F A117909 G.f.: (1 +2*x +x^3)/(1-x^3)^2.
%F A117909 a(n) = Sum_{k=0..n} 0^abs(L(C(n,2)/3) - 2*L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.
%F A117909 a(n) = sin((n+1)*Pi/3)*((8*n+14)*sin((n+1)*Pi/3) - sqrt(3)*cos(n*Pi))/9. - _Wesley Ivan Hurt_, Sep 24 2017
%t A117909 Riffle[Range[60],0,3] (* _Harvey P. Dale_, Sep 12 2013 *)
%o A117909 (Magma) I:=[1,2,0,3,4,0]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..91]]; // _G. C. Greubel_, Oct 21 2021
%o A117909 (Sage)
%o A117909 def a(n): return (2*n+3)/3 if (n%3==0) else 2*(n+2)/3 if (n%3==1) else 0
%o A117909 [a(n) for n in (0..90)] # _G. C. Greubel_, Oct 21 2021
%Y A117909 Cf. A009947, A117908.
%K A117909 easy,nonn
%O A117909 0,2
%A A117909 _Paul Barry_, Apr 01 2006