%I #3 Mar 30 2012 18:37:22
%S 1,2,3,6,11,18,28,42,61,88,124,168,227,302,394,510,653,826,1037,1290,
%T 1591,1952,2378,2876,3461,4144,4935,5850,6905,8114,9500,11080,12875,
%U 14914,17218,19816,22740,26024,29703,33812,38395,43500,49170,55460
%N Self-convolution of A180711.
%H Paul D. Hanna, <a href="/A180712/b180712.txt">Table of n, a(n), n = 0..500.</a>
%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 +...
%e The square-root of the g.f. begins:
%e sqrt(A(x)) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 9*x^9 + 11*x^10 + 12*x^11 + 15*x^12 + 17*x^13 + 19*x^14 +...
%e The integer part of the square-root of the coefficients of g.f. A(x) equals the respective coefficients of sqrt(A(x)), as illustrated by:
%e [sqrt(1)]=1; [sqrt(2)]=1; [sqrt(3)]=1; [sqrt(6)]=2; [sqrt(11)]=3;
%e [sqrt(18)]=4; [sqrt(28)]=5; [sqrt(42)]=6; [sqrt(61)]=7; [sqrt(88)]=9;
%e [sqrt(124)]=11; [sqrt(168)]=12; [sqrt(227)]=15; [sqrt(302)]=17; ...
%o (PARI) {a(n)=local(A=[1,1,1,t],T);for(i=1,n-2,for(N=A[ #A-1],2*A[ #A-1], T=subst(Vec(Ser(A)^2)[ #A],t,N);if(sqrtint(T)==N,A[ #A]=N;A=concat(A,t);break)));Vec(Ser(A)^2)[n+1]}
%Y Cf. A180711.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 22 2010