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

%I A385908 #15 Aug 01 2025 11:07:50
%S A385908 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,
%T A385908 112,119,180,167,271,267,374,380,536,519,804,768,1186,1137,1699,1632,
%U A385908 2442,2344,3594,3410,5281,5007,7597,7246,10992,10456,16124,15200,23460,22219,33908,32152,49284,46580
%N A385908 G.f. satisfies A(x) = A(x^2) - A(x^3)/A(-x^2).
%C A385908 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).
%H A385908 Paul D. Hanna, <a href="/A385908/b385908.txt">Table of n, a(n) for n = 1..3000</a>
%F A385908 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A385908 (1) A(x) = A(x^2) - A(x^3)/A(-x^2).
%F A385908 (2) A(x) = Sum_{n>=0} -A(x^(3*2^n)) / A(-x^(2*2^n)).
%e A385908 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 + ...
%e A385908 RELATED SERIES.
%e A385908 -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 + ...
%e A385908 SPECIFIC VALUES.
%e A385908 A(t) = 1 at t = 0.484687676611114689599533668921584333892926572607549... where 1 = A(t^2) - A(t^3)/A(-t^2).
%e A385908 A(t) = 2 at t = 0.626307557836743544400642615449370060133180589568562...
%e A385908 A(t) = 3 at t = 0.688425357998520764708465741188540670571098491468947...
%e A385908 A(t) = 4 at t = 0.722322807203734148016475080483297362895764436016224...
%e A385908 A(t) = 5 at t = 0.743435965558455262257890975727670149952734973906457... where 5 = A(t^2) - A(t^3)/A(-t^2).
%e A385908 A(1/2) = 1.07058481947548081707725269628863369127382368156458... where A(1/2) = A(1/4) - A(1/8)/A(-1/4).
%e A385908 A(1/3) = 0.51043013667634133844559016420320082768688442172785... where A(1/3) = A(1/9) - A(1/27)/A(-1/9).
%e A385908 A(1/4) = 0.33659795252020501354356922596423061291499721813786...
%e A385908 A(1/5) = 0.25136119039705534126580215871039458733095892423913...
%e A385908 A(1/8) = 0.14307514170085887429012626627006722045079984236860...
%e A385908 A(1/9) = 0.12513760312002194667097800612766717440651661319792...
%e A385908 A(1/16) = 0.06668103538984580162603374672451457685909316945786...
%e A385908 A(1/27) = 0.03846335312696051452727224567365991805112015192227...
%e A385908 A(1/64) = 0.01587307456112108371862387567387584552791396332498...
%e A385908 A(-1/4) = -0.19492874892212057242960516515401098780937269859766...
%e A385908 A(-1/9) = -0.09982896053535438726177476277299949942722497516842...
%e A385908 A(-1/16) = -0.0588072608781872517294441019994481590287607483365...
%o A385908 (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)}
%o A385908 for(n=1, 120, print1(a(n), ", "))
%K A385908 nonn
%O A385908 1,4
%A A385908 _Paul D. Hanna_, Jul 30 2025