A119471 G.f. A(x) equals the limit of the composition of functions F_n(x) = (x + x^n) starting with F_1(x) = x and repeating F_{n+1}(x) = F_n(x) + F_n(x)^(n+1). Thus, A(x) = ... o (x + x^n) o ... o (x + x^3) o (x + x^2) o x, where operator 'o' denotes composition.

1, 1, 1, 4, 8, 17, 50, 146, 399, 1087, 3042, 8741, 25509, 75259, 223529, 665215, 1983226, 5931158, 17800505, 53627756, 162206221, 492399027, 1499501067, 4579193127, 14017819056, 43001141630, 132154209754, 406818719006, 1254224223578, 3872184906554, 11970390368001, 37051341871259

Offset

1,4

Comments

Conjecture: the radius of convergence r satisfies A(r) = 1, where r = 0.3104429499 8756901289 5410716645 4712860635 3102024283 5871760958 3347115394 2327073684 7... Equivalently, if we set y = r and repeat y = y + y^n as n = 2,3,4,..., the process yields y = 1 as a limit.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1975

Example

G.f.: A(x) = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 50*x^7 + 146*x^8 + 399*x^9 + 1087*x^10 + 3042*x^11 + 8741*x^12 + ...
A(x) is the limit of the composition of functions F_n(x) = (x + x^n), starting with F_1(x) = x and repeating F_{n+1}(x) = (x + x^(n+1)) o F_n(x) as illustrated below.
F_2(x) = (x + x^2) o x;
F_3(x) = (x + x^3) o (x + x^2) o x = x + x^2 + x^3 + 3*x^4 + 3*x^5 + x^6;
F_4(x) = (x + x^4) o F_3(x) = x + x^2 + x^3 + 4*x^4 + 7*x^5 + 11*x^6 + ...
F_5(x) = (x + x^5) o F_4(x) = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 16*x^6 + ...
F_6(x) = (x + x^6) o F_5(x) = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 17*x^6 + ...
F_7(x) = (x + x^7) o (x + x^6) o (x + x^5) o (x + x^4) o (x + x^3) o (x + x^2) o x = x + x^2 + x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 50*x^7 + 145*x^8 + ...
etc.
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.288559191494800993203731731121982468098429942673212120699...;
  set y = t and repeat y = y + y^n as n = 2,3,4,... yields y = 1/2 as a limit.
A(t) = 2/5 at t = 0.26211206260215595456102382659148425321601535608074...
A(t) = 1/3 at t = 0.23648395533456266035083329395729431731120023685058...
A(t) = 1/4 at t = 0.19465116797744952166351583836381098310239167766184...
A(t) = 1/5 at t = 0.16414379034361087964647775235426196880343105426409...
A(2/7) = 0.48627618277358163879863010073510184673891142475353...
  set y = 2/7 and repeat y = y + y^n as n = 2,3,4,... yields y = A(2/7) as a limit.
A(1/4) = 0.36629298970934721419002882309226036579441425669523...
A(1/5) = 0.25952689303552916018870638690456573933139342060811...
A(1/6) = 0.20389303523340129652448502349743472476382872440444...

Prog

(PARI) {a(n) = my(F=x); for(k=2, n, F = subst(x + x^k, x, F +x*O(x^n)); ); polcoef(GF=F, n)}
{upto(n) = a(n); Vec(GF)}
upto(30) \\ program updated by Paul D. Hanna, Jul 09 2026

Crossrefs

Cf. A119470, A119472, A119459, A119460.

Sequence in context: A098125 A335274 A296399 * A145779 A252790 A144178

Adjacent sequences: A119468 A119469 A119470 * A119472 A119473 A119474

Keyword

nonn,changed

Author

Paul D. Hanna, May 22 2006

Extensions

Entry revised by Paul D. Hanna, Jul 09 2026

Status

approved