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A384833 G.f. satisfies A(x) = x + A(x^2)*A(x^3) with A(0) = 1.

%I A384833 #13 Jun 30 2025 10:03:58
%S A384833 1,1,1,1,1,1,2,1,2,2,2,2,4,2,3,4,4,3,7,4,6,6,7,5,12,6,9,11,11,8,18,10,
%T A384833 14,16,16,13,29,14,22,25,26,18,40,22,32,35,35,29,60,31,44,52,51,38,84,
%U A384833 44,66,71,71,55,118,59,88,101,98,75,158,84,121,132,131,102,222,109,163,183,183,132,288,149,220
%N A384833 G.f. satisfies A(x) = x + A(x^2)*A(x^3) with A(0) = 1.
%H A384833 Paul D. Hanna, <a href="/A384833/b384833.txt">Table of n, a(n) for n = 0..4100</a>
%F A384833 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A384833 (1) A(x) = x + A(x^2)*A(x^3).
%F A384833 (2) A(x) = x + (x^2 + A(x^4)*A(x^6)) * A(x^3).
%F A384833 (3) A(x) = x + (x^3 + A(x^6)*A(x^9)) * A(x^2).
%F A384833 (4) A(x) = x + (x^2 + A(x^4)*A(x^6)) * (x^3 + A(x^6)*A(x^9)).
%e A384833 G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 4*x^12 + 2*x^13 + 3*x^14 + 4*x^15 + 4*x^16 + 3*x^17 + 7*x^18 + ...
%e A384833 SPECIFIC VALUES.
%e A384833 A(t) = 8 at t = 0.815567952503420060983324003731884365610153065333843...
%e A384833 A(t) = 7 at t = 0.801345796531134685781199627451222676066060436869548...
%e A384833 A(t) = 6 at t = 0.782464582280003562207635790317752834218000582442380...
%e A384833 A(t) = 5 at t = 0.755941352922813739702152230754990246964815214312332...
%e A384833 A(t) = 4 at t = 0.715501328653246854491171663936314129852235838729017...
%e A384833 A(t) = 3 at t = 0.645441642217772666989026706116848414808373572998316...
%e A384833 A(t) = 2 at t = 0.494376497811120189544751969812464177694437314637999...
%e A384833   where 2 = t + A(t^2)*A(t^3).
%e A384833 A(1/2) = 2.02411711463174118628591070519235301076213003753496853...
%e A384833   where A(1/2) = 1/2 + A(1/4)*A(1/8).
%e A384833 A(1/3) = 1.50160454820609473881444517916924635584141796658988569...
%e A384833   where A(1/3) = 1/3 + A(1/9)*A(1/27).
%e A384833 A(1/4) = 1.33359794441646176628139897881393129146941143225096194...
%e A384833   where A(1/4) = 1/4 + A(1/16)*A(1/64).
%e A384833 A(1/5) = 1.25006720843726236545112043092498433427969081536440717...
%e A384833 A(1/6) = 1.20002214885614821249840694253067786363033380936497161...
%e A384833 A(1/8) = 1.14286102570339840364008696181046539894860083304162542...
%e A384833 A(1/9) = 1.12500190781795393807826165839192346764220418556447137...
%e A384833 A(1/16) = 1.0666667265196712497741852980367608423574849397188049...
%e A384833 A(1/27) = 1.0384615410463901537541097310422519024141653373409764...
%e A384833 A(1/64) = 1.0158730158875713973506228538750118258791597258893753...
%t A384833 terms = 81; A[_] = 1; Do[A[x_] = x + A[x^2] * A[x^3] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Stefano Spezia_, Jun 29 2025 *)
%o A384833 (PARI) {a(n) = my(A=1+x +x*O(x^n)); for(i=1, ceil(log(n+2)/log(2)), A = x + subst(A, x, x^2)*subst(A, x, x^3) +x*O(x^n); ); polcoef(A, n)}
%o A384833 for(n=0, 80, print1(a(n), ", "))
%Y A384833 Cf. A382126.
%K A384833 nonn
%O A384833 0,7
%A A384833 _Paul D. Hanna_, Jun 29 2025