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%I #15 Mar 24 2023 10:40:10
%S 1,2,4,7,12,18,25,34,44,55,68,82,97,114,132,151,172,194,217,242,268,
%T 295,324,354,385,418,452,487,524,562,601,642,684,727,772,818,865,914,
%U 964,1015,1068,1122,1177,1234,1292,1351,1412,1474,1537,1602,1668,1735,1804,1874
%N Expansion of g.f.: (1+x^2+2*x^4)/((1-x^3)*(1-x)^2).
%H G. C. Greubel, <a href="/A084672/b084672.txt">Table of n, a(n) for n = 0..5000</a>
%H J. Hietarinta and C.-M. Viallet, <a href="https://doi.org/10.1016/S0960-0779(98)00266-5">Discrete Painlevé I and singularity confinement in projective space</a>, Chaos, Solitons and Fractals 11, 2000, p. 29.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F From _G. C. Greubel_, Mar 22 2023: (Start)
%F a(n) = (1/3)*(2*n^2 - n + 3 + 2*floor((n+2)/3) + floor((n+1)/3)).
%F a(n) = (1/9)*(6*n^2 + 11 - 2*A049347(n) - A049347(n-1)). (End)
%t CoefficientList[Series[(1+x^2+2x^4)/((1-x^3)(1-x)^2), {x,0,70}], x] (* _Harvey P. Dale_, Mar 31 2011 *)
%o (PARI) Vec((1+x^2+2*x^4)/(1-x^3)/(1-x)^2+O(x^99)) \\ _Charles R Greathouse IV_, Sep 10 2014
%o (Magma) [(2*n^2 -n +3 +2*Floor((n+2)/3) +Floor((n+1)/3))/3: n in [0..60]]; // _G. C. Greubel_, Mar 22 2023
%o (SageMath) [(2*n^2 -n +3 +2*((n+2)//3) +((n+1)//3))/3 for n in range(61)] # _G. C. Greubel_, Mar 22 2023
%Y Cf. A049347.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jul 16 2003
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