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A384267 G.f. A(x) satisfies A(x) = 1 + abs( x/A(x)^2 ).

%I A384267 #10 Jun 20 2025 10:46:39
%S A384267 1,1,2,1,6,13,4,80,242,109,1702,5177,2208,40348,128560,56864,1052102,
%T A384267 3406333,1509862,28900645,94971462,42420281,825816148,2740269448,
%U A384267 1228678588,24277298940,81183221736,36526643608,729682028652,2454721201940,1107304048024,22319301025880,75450489469554
%N A384267 G.f. A(x) satisfies A(x) = 1 + abs( x/A(x)^2 ).
%C A384267 Conjectures:
%C A384267 (C.1) a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) (cf. A006013).
%C A384267 (C.2) [x^(3*n+1)] x/A(x)^2 > 0, [x^(3*n+2)] x/A(x)^2 < 0, and [x^(3*n+3)] x/A(x)^2 < 0 for n >= 0.
%C A384267 (C.3) The values of a(n)/a(n-1) tend to a period-3 sequence of reals near [21.83826..., 3.53749..., 0.46127...] (the values at n = 5002..5004).
%H A384267 Paul D. Hanna, <a href="/A384267/b384267.txt">Table of n, a(n) for n = 0..1000</a>
%F A384267 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A384267 (1) A(x) = 1 + x * abs( 1/A(x)^2 ).
%F A384267 (2) A(x) = 1 + x * ( 2/A(w*x)^2 + 2/A(w^2*x)^2 - 1/A(x)^2 )/3, where w = exp(i*2*Pi/3) = -1/2 + sqrt(3)/2*i (conjecture); this is implied by conjecture (C.2).
%e A384267 G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 6*x^4 + 13*x^5 + 4*x^6 + 80*x^7 + 242*x^8 + 109*x^9 + 1702*x^10 + 5177*x^11 + 2208*x^12 + ...
%e A384267 RELATED SERIES.
%e A384267 A(x) equals the series formed from the absolute values of the coefficients in 1 + x/A(x)^2 where
%e A384267 x/A(x)^2 = x - 2*x^2 - x^3 + 6*x^4 - 13*x^5 - 4*x^6 + 80*x^7 - 242*x^8 - 109*x^9 + 1702*x^10 - 5177*x^11 - 2208*x^12 +-- ...
%e A384267 notice that the signs in x/A(x)^2 seem to be {+,-,-} repeating.
%e A384267 SPECIFIC VALUES.
%e A384267 A(t) = 17/10 at t = 0.2988099109194334966744754680560903...
%e A384267 A(t) = 8/5 at t = 0.28632536002959676347841744332637502584281553236328...
%e A384267 A(t) = 3/2 at t = 0.26584952269781748463288503061262604182943155912168...
%e A384267 A(t) = 7/5 at t = 0.23679807527229400928334910529482907166586736528066...
%e A384267 A(t) = 4/3 at t = 0.21222131698512068142939257924460486238379301612052...
%e A384267 A(t) = 6/5 at t = 0.14851037601497632663099987292554419705752970437155...
%e A384267 A(1/4) = 1.4416840609369316144418746432100574811353758654573...
%e A384267 A(1/5) = 1.3040757997934088091953759590684948311334157108446...
%e A384267 A(1/6) = 1.2337286609104904159907289378298492254783023920577...
%e A384267 A(1/7) = 1.1900466603567900992777823995090950832516801703123...
%e A384267 A(1/8) = 1.1601356692672906064760109443886299674930778512606...
%e A384267 A(1/9) = 1.1383397014975021472515053785203604745989973570682...
%e A384267 A(1/10) = 1.1217447587000441822177506555087189442697776256039...
%o A384267 (PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + x*Ser(abs(Vec(1/(A +x*O(x^n))^2))) ); polcoef(A, n)}
%o A384267 for(n=0, 32, print1(a(n), ", "))
%Y A384267 Cf. A380708, A380710, A006013.
%K A384267 nonn
%O A384267 0,3
%A A384267 _Paul D. Hanna_, Jun 19 2025