This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309303 #21 Jan 30 2020 21:29:18
%S A309303 0,1,-1,2,-1,4,2,13,23,68,164,439,1146,3067,8231,22306,60791,166684,
%T A309303 459308,1271479,3534116,9859573,27598757,77490472,218183522,615902899,
%U A309303 1742738477,4942022648,14043034703,39979680748,114020882010,325721340061
%N A309303 Expansion of g.f. (sqrt(x+1) - sqrt(1-3*x))/(2*(x+1)^(3/2)).
%C A309303 (-1)^a(n) = (-1)^A010060(n) = A106400(n) (Thue-Morse sequence).
%C A309303 a(n) + a(n+1) = A005043(n) = (-1)^n * hypergeom([-n, 1/2], [2], 4) (Motzkin sums).
%H A309303 Robert Israel, <a href="/A309303/b309303.txt">Table of n, a(n) for n = 0..2106</a>
%H A309303 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Thue-MorseSequence.html">Thue-Morse sequence</a>.
%H A309303 Wikipedia, <a href="https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence">Thue-Morse sequence</a>.
%F A309303 a(n) = (-1)^n/2 + 3^(n+3/2)/2^(n+4) * (2*n-3)!!/n! * hypergeom([3/2, 3/2], [3/2 - n], 1/4).
%F A309303 D-finite with recurrence: n*a(n) = (n-4)*a(n-1) + (n-2)*(5*a(n-2) + 3*a(n-3)).
%F A309303 a(n) ~ c * 3^n / n^(3/2), where c = 3^(3/2) / (32*sqrt(Pi)) = 0.09161297...
%p A309303 f:= gfun:-rectoproc({n*a(n) = (n-4)*a(n-1) + (n-2)*(5*a(n-2) + 3*a(n-3)),a(0)=0,a(1)=1,a(2)=-1},a(n),remember):
%p A309303 map(f, [$0..40]); # _Robert Israel_, Jul 23 2019
%t A309303 Table[(-1)^n/2 + 3^(n + 3/2)/2^(n + 4) (2 n - 3)!!/n! Hypergeometric2F1[3/2, 3/2, 3/2 - n, 1/4], {n, 0, 31}]
%Y A309303 Cf. A005043, A010060, A106400.
%K A309303 sign
%O A309303 0,4
%A A309303 _Vladimir Reshetnikov_, Jul 21 2019