A385908 G.f. satisfies A(x) = A(x^2) - A(x^3)/A(-x^2).

1, 1, 1, 2, 0, 2, 2, 2, 4, 3, 4, 7, 4, 6, 8, 9, 14, 17, 19, 21, 24, 28, 36, 42, 63, 58, 85, 88, 112, 119, 180, 167, 271, 267, 374, 380, 536, 519, 804, 768, 1186, 1137, 1699, 1632, 2442, 2344, 3594, 3410, 5281, 5007, 7597, 7246, 10992, 10456, 16124, 15200, 23460, 22219, 33908, 32152, 49284, 46580

Offset: 1

Author

Paul D. Hanna, Jul 30 2025

Keywords

• nonn

Comments

a(n) ~ c*d^n where d = 1.205173231150326066922845793121761178534075668579830253410... and c = 0.564131292071378563944161433232744388888126640604598... if n is odd, and c = 0.441434491751921785708450528534195744698243710446358... if n is even. The radius of convergence r = 1/d = 0.829756232674957256030569741672368872941406926765977... satisfies A(-r^2) = 0 and A(r^4)*A(-r^4) = A(-r^6).

Examples

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 4*x^9 + 3*x^10 + 4*x^11 + 7*x^12 + 4*x^13 + 6*x^14 + 8*x^15 + 9*x^16 + 14*x^17 + 17*x^18 + ...
RELATED SERIES.
-A(x^3)/A(-x^2) = x + x^3 + x^4 + x^6 + 2*x^7 + 4*x^9 + 3*x^10 + 4*x^11 + 5*x^12 + 4*x^13 + 4*x^14 + 8*x^15 + 7*x^16 + 14*x^17 + 13*x^18 + ...
SPECIFIC VALUES.
A(t) = 1 at t = 0.484687676611114689599533668921584333892926572607549... where 1 = A(t^2) - A(t^3)/A(-t^2).
A(t) = 2 at t = 0.626307557836743544400642615449370060133180589568562...
A(t) = 3 at t = 0.688425357998520764708465741188540670571098491468947...
A(t) = 4 at t = 0.722322807203734148016475080483297362895764436016224...
A(t) = 5 at t = 0.743435965558455262257890975727670149952734973906457... where 5 = A(t^2) - A(t^3)/A(-t^2).
A(1/2) = 1.07058481947548081707725269628863369127382368156458... where A(1/2) = A(1/4) - A(1/8)/A(-1/4).
A(1/3) = 0.51043013667634133844559016420320082768688442172785... where A(1/3) = A(1/9) - A(1/27)/A(-1/9).
A(1/4) = 0.33659795252020501354356922596423061291499721813786...
A(1/5) = 0.25136119039705534126580215871039458733095892423913...
A(1/8) = 0.14307514170085887429012626627006722045079984236860...
A(1/9) = 0.12513760312002194667097800612766717440651661319792...
A(1/16) = 0.06668103538984580162603374672451457685909316945786...
A(1/27) = 0.03846335312696051452727224567365991805112015192227...
A(1/64) = 0.01587307456112108371862387567387584552791396332498...
A(-1/4) = -0.19492874892212057242960516515401098780937269859766...
A(-1/9) = -0.09982896053535438726177476277299949942722497516842...
A(-1/16) = -0.0588072608781872517294441019994481590287607483365...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..3000

Programs

PARI

{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, ceil(log(n+2)/log(2)), A = subst(A, x, x^2) - subst(A, x, x^3)/subst(A, x, -x^2) +x*O(x^n); ); polcoef(A, n)}
for(n=1, 120, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = A(x^2) - A(x^3)/A(-x^2).

(2) A(x) = Sum_{n>=0} -A(x^(3*2^n)) / A(-x^(2*2^n)).