A396100 - OEIS

%I #20 Jun 04 2026 11:31:44

%S 1,1,2,3,6,12,17,48,52,200,162,848,502,3631,1605,15730,5063,68501,

%T 16314,300199,52203,1320571,169128,5831378,547153,25825144,1782944,

%U 114682172,5817259,510476945,19066119,2277257478,62627721,10179548297,206383073,45590331780,681724709,204548718149,2257730658

%N G.f. satisfies A(x) = (1/x)*A(A(x^2)) + A(A(A(x^2))).

%H Paul D. Hanna, <a href="/A396100/b396100.txt">Table of n, a(n) for n = 1..4000</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n has the following properties.

%F (1) A(x) = (1/x)*A(A(x^2)) + A(A(A(x^2))).

%F (2) A(A(x^2)) = x*(A(x) - A(-x))/2.

%F (3) A(A(A(x^2))) = (A(x) + A(-x))/2.

%F (4) A(A(x^2)) * A(A(A(x^2))) = x*(A(x)^2 - A(-x)^2)/4.

%F (5) A(x) converges for |x| < r where r = A(A(r^2)) at r = 0.460416184803533005220454820697720660656558236530176031947023...

%F (6) A(A(x)) converges for |x| < A(r^2) where A(r^2) = 0.286307088477904580502293384159028170286185128654277103146166...

%F (7) A(A(A(x))) converges for |x| < r^2 where r^2 = 0.211983063229041056608973065058713002287454641858749290265267...

%F (8) A(A(A(A(x)))) converges for |x| < q where A(q) = r^2 at q = 0.169686836320072198239498124401300712862395294961628283115207...

%e G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 17*x^7 + 48*x^8 + 52*x^9 + 200*x^10 + 162*x^11 + 848*x^12 + ...

%e where A(x) = A(A(x^2))/x + A(A(A(x^2))).

%e RELATED SERIES.

%e The second and third iterations of A(x) begin

%e A(A(x)) = x + 2*x^2 + 6*x^3 + 17*x^4 + 52*x^5 + 162*x^6 + 502*x^7 + 1605*x^8 + 5063*x^9 + 16314*x^10 + ...

%e A(A(A(x))) = x + 3*x^2 + 12*x^3 + 48*x^4 + 200*x^5 + 848*x^6 + 3631*x^7 + 15730*x^8 + 68501*x^9 + 300199*x^10 + ...

%e the coefficients of which interleave to form this sequence.

%e SPECIFIC VALUES.

%e A(t) = 1 at t = 0.39673778500057382387495833993119539750836036095156...

%e A(A(t)) = 1 at t = 0.26255360314235395678135388941463883684788790226647...

%e A(A(A(t))) = 1 at t = 0.19926900486604179799565952253637071697535783657489...

%e A(t) = 2 at t = 0.44875853546126801043241091687812220875546168205825...

%e A(A(t)) = 2 at t = 0.28221464120396115535295360031881516608029031654276...

%e A(A(A(t))) = 2 at t = 0.20984343720647621630080412125020563105221221948719...

%e A(1/3) = 0.62229409951352684257488032885327282020663717653567...

%e A(1/4) = 0.36653400632354995092423995727875573563570860963597...

%e A(1/5) = 0.26388356638522249566069381720667867660165153315328...

%e A(A(1/9)) = 0.14797203450652118718881273999272077710797258706667...

%e A(A(A(1/9))) = 0.17837799599396328100844210887511048888271941533564...

%e   where A(1/3) = 3*A(A(1/9)) + A(A(A(1/9))).

%e A(A(1/16)) = 0.0720983262045680584895309647699304129073555208871...

%e A(A(A(1/16))) = 0.0781407015052777169661160981990340840062865260875...

%e   where A(1/4) = 4*A(A(1/16)) + A(A(A(1/16))).

%o (PARI) {a(n) = my(A = x,B,C,N=ceil(log(n)/log(2))); for(k=1,N,

%o A = truncate(A) + x*O(x^min(n,2^k)); B = subst(A,x,A); C = subst(A,x,B);

%o A = subst(B,x,x^2)/x + subst(C,x,x^2); ); polcoef(GF=A +x*O(x^n),n)}

%o {upto(n) = a(n); Vec(GF)}

%o upto(40)

%Y Cf. A120566.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jun 01 2026