A180711 - OEIS

%I #3 Mar 30 2012 18:37:22

%S 1,1,1,2,3,4,5,6,7,9,11,12,15,17,19,22,25,28,32,35,39,44,48,53,58,64,

%T 70,76,83,90,97,105,113,122,131,140,150,161,172,183,195,208,221,235,

%U 249,264,280,296,313,331,349,368,387,408,429,451,474,497,522,547,573,600

%N a(n) = [x^n] A(x) = floor(sqrt( b(n) )) where b(n) = [x^n] A(x)^2.

%H Paul D. Hanna, <a href="/A180711/b180711.txt">Table of n, a(n), n = 0..500.</a>

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 +...

%e The square of the g.f. A(x) begins:

%e A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 42*x^7 + 61*x^8 + 88*x^9 + 124*x^10 + 168*x^11 + 227*x^12 + 302*x^13 + 394*x^14 +...

%e The integer part of the square-root of the coefficients of the square of the g.f. A(x) equals the respective coefficients of 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)));A[n+1]}

%Y Cf. A180712.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 22 2010