A124571 Expansion of limit b(n)/x^n where b(n) = b(n-1)^2 + b(n-1)*x, b(1) = x^2.

1, 1, 1, 3, 4, 6, 12, 21, 33, 53, 90, 151, 253, 426, 701, 1151, 1900, 3123, 5162, 8553, 14092, 23223, 38296, 62963, 103458, 170056, 279140, 457833, 751033, 1231671, 2019090, 3309710, 5424315, 8886249, 14553015, 23826952, 38997232, 63806394

Offset

1,4

Comments

What is the limit of a(n)^(1/n)? For example, a(2000)^(1/2000) = 1.6192745710... a(3000)^(1/3000) = 1.6189671448... - Paul D. Hanna, Jul 28 2025

Conjecture: the limit is equal to GoldenRatio. - Vaclav Kotesovec, Jul 29 2025

Links

Paul D. Hanna, Table of n, a(n) for n = 1..2000

Formula

G.f. A(x) = lim_{n->oo} S(n) where S(n+1) = S(n)*(1 + x^n*S(n)), with S(0) = x. - Paul D. Hanna, Jul 28 2025

Example

G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 12*x^7 + 21*x^8 + 33*x^9 + 53*x^10 + 90*x^11 + 151*x^12 + ...
G.f. A(x) equals the limit of S(n) starting with S(0) = x, S(1) = x*(1+x), S(2) = S(1)*(1 + x*S(1)), S(3) = S(2)*(1 + x^2*S(2)), S(4) = S(3)*(1 + x^3*S(3)), etc. - Paul D. Hanna, Jul 28 2025

Mathematica

nmax = 50; s = x; Do[s = Normal[s*(1 + x^k*s) + O[x]^(nmax + 1)]; , {k, 0, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 29 2025 *)

Prog

(PARI) {a(n) = my(A); if(n<1, 0, A=x+x*O(x^n); for(k=0, n-2, A+=A^2*x^k); polcoeff(A, n))}
(PARI) N = 50; \\ N = number of terms
{my(S=x); for(n=0, N, S = S*(1 + x^n*S) +x*O(x^N)); Vec(S)} \\ Paul D. Hanna, Jul 28 2025

Crossrefs

Cf. A001622.

Sequence in context: A109069 A329953 A329671 * A332279 A095765 A095016

Adjacent sequences: A124568 A124569 A124570 * A124572 A124573 A124574

Keyword

nonn

Author

Michael Somos, Nov 04 2006

Status

approved