OFFSET
2,2
COMMENTS
It appears that a(2*n) == 1 (mod 8) and a(2*n+1) == 3 (mod 8) for n >= 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 2..410
FORMULA
G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.
(1) A(x - A(x)) = x^2 + x*A(x).
(2) A(x - x^2 - (1+x)*A(x)) = (x - A(x)) * (x + x^2 - (1-x)*A(x)).
(3) A(x) = 2*B(x)^2 - x*B(x) where B(x) = x + A(B(x)) and B(x - A(x)) = x.
(4) A(x) = x - Series_Reversion( (x + sqrt(x^2 + 8*A(x)))/4 ).
EXAMPLE
G.f.: A(x) = x^2 + 3*x^3 + 17*x^4 + 131*x^5 + 1209*x^6 + 12603*x^7 + 143705*x^8 + 1757491*x^9 + 22757185*x^10 + ...
where A(x - A(x)) = x^2 + x*A(x).
RELATED SERIES.
A(x - A(x)) = x^2 + x^3 + 3*x^4 + 17*x^5 + 131*x^6 + 1209*x^7 + 12603*x^8 + ...
Let B(x) satisfy B(x - A(x)) = x then B(x) = x + A(B(x)) where
B(x) = x + x^2 + 5*x^3 + 37*x^4 + 337*x^5 + 3505*x^6 + 40069*x^7 + 492413*x^8 + 6413425*x^9 + 87681121*x^10 + ...
B(x)^2 = x^2 + 2*x^3 + 11*x^4 + 84*x^5 + 773*x^6 + 8054*x^7 + 91887*x^8 + 1124952*x^9 + 14585305*x^10 + ...
where A(x) = 2*B(x)^2 - x*B(x).
PROG
(PARI) {a(n) = my(V=[0, 0, 1], A=x); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef(x^2 + x*A - subst(A, x, x - A), #V-1) ); V[n+1]}
for(n=2, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 03 2026
STATUS
approved