OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..600
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( (x - A(x))*(A(x) - 2*x) ).
(2) A(x)^9 = A( ((x - A(x))*(A(x) - 2*x) - A(x)^3) * (A(x)^3 - 2*(x - A(x))*(A(x) - 2*x)) ).
(3) x^3 = A( (B(x) - x)*(x - 2*B(x)) ) where A(B(x)) = x.
EXAMPLE
G.f.: A(x) = x + x^2 + 4*x^3 + 23*x^4 + 155*x^5 + 1151*x^6 + 9084*x^7 + 74772*x^8 + 634752*x^9 + 5517179*x^10 + 48856250*x^11 + 439210124*x^12 + ...
where A(x)^3 = A( (x - A(x))*(A(x) - 2*x) ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 15*x^5 + 94*x^6 + 663*x^7 + 5052*x^8 + 40546*x^9 + 337698*x^10 + 2891871*x^11 + 25304819*x^12 + ...
(x - A(x))*(A(x) - 2*x) = x^3 + 3*x^4 + 15*x^5 + 93*x^6 + 657*x^7 + 5013*x^8 + 40266*x^9 + 335565*x^10 + 2874825*x^11 + 25163523*x^12 + ...
SPECIFIC VALUES.
A(1/11) = 0.107841209855945835597327800288172120205254638166959...
where A(1/11)^3 = A( (1/11 - A(1/11))*(A(1/11) - 2/11) ).
A(1/12) = 0.095638462555899438936665428468730427815713586552270...
where A(1/12)^3 = A( (1/12 - A(1/12))*(A(1/12) - 2/12) ).
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( Ax^3 - subst(Ax, x, (x - Ax)*(Ax - 2*x) ), #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 19 2024
STATUS
approved