A397525 - OEIS

%I #25 Jul 13 2026 22:16:10

%S 2,1,3,5,27,39,99,161,1699,2523,6069,10603,52223,84083,195171,342329,

%T 6714211,10745171,24449313,44288583,207189205,354267801,797990165,

%U 1417980247,13384760583,22466908431,50281659209,91305188583,418942004743,731825520779,1621903512419

%N a(n) = 2^(A007238(n)) * [x^n] B(x) where B(x) satisfies (B(x)^2-B(x^2))/2 = 1/(1-x)^2 with a(0) = 2.

%C This sequence was inspired by the analysis of A397174 and A394611. The goal was to understand the theoretical foundation of the 1/sqrt(2) growth observed in A394611 and also to understand its fluctuations better.

%C A(x) in the name of this sequence has the infinite nested square root representation: A(x) = sqrt(2/(1-x)^2 + sqrt(2/(1-x^(2^1))^2 + sqrt(2/(1-x^(2^2))^2 + sqrt(2/(1-x^(2^3))^2 + ...)))).

%C A397521 shows a surprisingly different behavior while the definition differs only in replacing 1/(1-x)^2 by x/(1-x)^2.

%F sqrt(2) = lim_{m->oo} (1/m)*Sum_{k=0..m} a(k)*2^(-A007238(k)), with A007238(0) = 0.

%F Sum_{k=0..n} a(k)*a(n-k)*2^(-A007238(k)-A007238(n-k)) = 1 + 2*n + [n even] a(n/2)*2^(-A007238(n/2)), with A007238(0) = 0.

%o (PARI)

%o h(v) = (Ser(v)^2-subst(Ser(v),x,x^2))/2

%o ListA(max_n) = {my(v=[2]); while(#v<max_n, v=concat(v, ((#v+1)-Vec(Vec(h(concat(v, y)))[#v+1])[2])/2)); vector(#v, k, numerator(v[k]))}

%Y Cf. A007238, A394611, A397173, A397174, A397521.

%K nonn,new

%O 0,1

%A _Thomas Scheuerle_, Jun 29 2026