OFFSET
1,3
COMMENTS
The limit lim_{n->oo} a(n)^(1/n^2) = d appears to be near sqrt(2).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..100
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n along with B(x) = x + A(x) satisfies the following formulas.
(1) A(x) = x * sqrt( 1 + A(B(x)) ).
(2) A(x) = x * sqrt( 1 + B(x) * sqrt( 1 + A( B(B(x)) ) ) ).
(3) A(x) = x * sqrt( 1 + B(x) * sqrt( 1 + B(B(x)) * sqrt(1 + A( B(B(B(x))) ) ) ) ).
(4) B^n(x) = x + Sum_{k=0..n-1} A(B^k(x)) for n > 0, where B^n(x) = B( B^{n-1}(x) ) with B^0(x) = x.
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 9*x^4 + 82*x^5 + 1447*x^6 + 49360*x^7 + 3285360*x^8 + 430454058*x^9 + 111704560283*x^10 + ...
where A(x)^2 = x^2 + x^2*A(x + A(x)).
RELATED SERIES.
Let B(x) = x + A(x), then A(x) = x*sqrt(1 + A(B(x))) may be defined as the infinitely nested radical involving iterations of B(x) illustrated by
A(x) = x*sqrt(1 + B(x)*sqrt(1 + B^2(x)*sqrt(1 + B^3(x)*sqrt(1 + B^4(x)*sqrt(1 + ...))))).
The iterations of B(x) may be written like so
B^2(x) = B(B(x)) = x + A(x) + A(B(x)),
B^3(x) = B(B(B(x))) = x + A(x) + A(B(x)) + A(B(B(x))),
etc.
Thus, for n > 0, the n-th iteration of B(x) = x + A(x) may be expressed as
B^n(x) = x + A(x) + A(B(x)) + A(B^2(x)) + A(B^3(x)) + ... + A(B^{n-1}(x)),
where the iterations of B(x) as power series begin
B(x) = 2*x + x^2 + 2*x^3 + 9*x^4 + 82*x^5 + 1447*x^6 + ...
B^2(x) = 4*x + 6*x^2 + 24*x^3 + 195*x^4 + 3176*x^5 + 103470*x^6 + ...
B^3(x) = 8*x + 28*x^2 + 224*x^3 + 3498*x^4 + 109160*x^5 + 6904400*x^6 + ...
B^4(x) = 16*x + 120*x^2 + 1920*x^3 + 58980*x^4 + 3613552*x^5 + 450973392*x^6 + ...
B^5(x) = 32*x + 496*x^2 + 15872*x^3 + 967944*x^4 + 117584736*x^5 + ...
etc.
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = x*sqrt(1 + subst(A, x, x + A + x*O(x^n))) ); polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 12 2025
STATUS
approved