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a(n) = 2^(n - 1) + binomial(n, floor(n/2))*(n + 1)/2.
2

%I #18 Nov 17 2023 09:56:05

%S 1,2,5,10,23,46,102,204,443,886,1898,3796,8054,16108,33932,67864,

%T 142163,284326,592962,1185924,2464226,4928452,10209620,20419240,

%U 42190558,84381116,173962532,347925064,715908428,1431816856,2941192472,5882384944,12065310083,24130620166

%N a(n) = 2^(n - 1) + binomial(n, floor(n/2))*(n + 1)/2.

%C This sequence was obtained by omitting the two initial zeros from A191386, which has a more complicated definition. The simple formula defining this sequence was found by Dan Velleman and _Stan Wagon_. See A191386 for further information, including references.

%H Winston de Greef, <a href="/A323988/b323988.txt">Table of n, a(n) for n = 0..3300</a>

%F G.f. = (1+s)/(2*s*(1-2*x), where s = sqrt(1-4*x^2).

%F a(0) = 1, a(1) = 2, a(2) = 5; thereafter (8*n+16)*a(n) + (-4*n-8)*a(n+1) + (-2*n-6)*a(n+2) + (n+3)*a(n+3) = 0.

%t A323988[n_]:=2^(n-1)+Binomial[n,Floor[n/2]](n+1)/2;Array[A323988,50,0] (* _Paolo Xausa_, Nov 17 2023 *)

%o (PARI) a(n) = 2^(n-1) + binomial(n, n\2)*(n+1)/2 \\ _Winston de Greef_, Sep 17 2023

%Y Cf. A191386.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Feb 13 2019