%I #15 Sep 26 2024 03:10:51

%S 1,2,4,6,9,14,23,40,73,138,267,524,1037,2062,4111,8208,16401,32786,

%T 65555,131092,262165,524310,1048599,2097176,4194329,8388634,16777243,

%U 33554460,67108893,134217758,268435487,536870944,1073741857,2147483682,4294967331,8589934628

%N a(n) = [x^n] (1 - 2*x + x^2 - 2*x^3 + x^4)/((1 - x)^2*(1 - 2*x)).

%H G. C. Greubel, <a href="/A323227/b323227.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).

%F a(n) = Sum_{k=0..n} ( binomial(n - 2, k - 1) + 1 ), if n >= 2.

%F a(n) = ((2 - 2*n)*a(n-2) - (5 - 3*n)*a(n-1))/(n - 2) for n >= 4.

%F a(n+1) - (n + 1) = A094373(n) for n >= 0.

%F a(n+1) - a(n) = 2^n + 1 for n >= 2.

%F a(n) = A270841(n) = 2^(n-2)+n+1 for n>=2. - _R. J. Mathar_, Feb 14 2019

%F E.g.f.: (1/4)*(-(1 + 2*x) + 4*(1+x)*exp(x) + exp(2*x)). - _G. C. Greubel_, Sep 26 2024

%p a := proc(n) option remember; if n < 4 then return [1, 2, 4, 6][n + 1] fi;

%p ((2 - 2*n)*a(n-2) - (5 - 3*n)*a(n-1))/(n - 2) end: seq(a(n), n=0..35);

%t A323211[n_, k_] := If[n <= 1, 1, Binomial[n - 2, k - 1] + 1];

%t Table[Sum[A323211[n, k], {k, 0, n}], {n, 0, 35}]

%o (Magma)

%o [n le 1 select n+1 else 2^(n-2) +(n+1) : n in [0..35]]; // _G. C. Greubel_, Sep 26 2024

%o (SageMath)

%o [2^(n-2) +(n+1) -int(n==0)/4 -int(n==1)/2 for n in range(36)] # _G. C. Greubel_, Sep 26 2024

%Y Cf. A000051, A094373, A270841, A323211.

%K nonn,easy

%O 0,2

%A _Peter Luschny_, Feb 12 2019