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

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, 14, 16, 16, 13, 29, 14, 22, 25, 26, 18, 40, 22, 32, 35, 35, 29, 60, 31, 44, 52, 51, 38, 84, 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

Offset: 0

Author

Paul D. Hanna, Jun 29 2025

Keywords

• nonn

Examples

			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 + ...
SPECIFIC VALUES.
A(t) = 8 at t = 0.815567952503420060983324003731884365610153065333843...
A(t) = 7 at t = 0.801345796531134685781199627451222676066060436869548...
A(t) = 6 at t = 0.782464582280003562207635790317752834218000582442380...
A(t) = 5 at t = 0.755941352922813739702152230754990246964815214312332...
A(t) = 4 at t = 0.715501328653246854491171663936314129852235838729017...
A(t) = 3 at t = 0.645441642217772666989026706116848414808373572998316...
A(t) = 2 at t = 0.494376497811120189544751969812464177694437314637999...
  where 2 = t + A(t^2)*A(t^3).
A(1/2) = 2.02411711463174118628591070519235301076213003753496853...
  where A(1/2) = 1/2 + A(1/4)*A(1/8).
A(1/3) = 1.50160454820609473881444517916924635584141796658988569...
  where A(1/3) = 1/3 + A(1/9)*A(1/27).
A(1/4) = 1.33359794441646176628139897881393129146941143225096194...
  where A(1/4) = 1/4 + A(1/16)*A(1/64).
A(1/5) = 1.25006720843726236545112043092498433427969081536440717...
A(1/6) = 1.20002214885614821249840694253067786363033380936497161...
A(1/8) = 1.14286102570339840364008696181046539894860083304162542...
A(1/9) = 1.12500190781795393807826165839192346764220418556447137...
A(1/16) = 1.0666667265196712497741852980367608423574849397188049...
A(1/27) = 1.0384615410463901537541097310422519024141653373409764...
A(1/64) = 1.0158730158875713973506228538750118258791597258893753...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..4100

Crossrefs

Cf. A382126.

Programs

Mathematica

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 *)

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)}
for(n=0, 80, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = x + A(x^2)*A(x^3).

(2) A(x) = x + (x^2 + A(x^4)*A(x^6)) * A(x^3).

(3) A(x) = x + (x^3 + A(x^6)*A(x^9)) * A(x^2).

(4) A(x) = x + (x^2 + A(x^4)*A(x^6)) * (x^3 + A(x^6)*A(x^9)).