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%I #13 Nov 25 2024 07:13:39
%S 1,1,1,4,10,28,82,256,808,2617,8566,28447,95416,323002,1101724,
%T 3783769,13071556,45396925,158400937,555035695,1952238598,6890367190,
%U 24395779840,86623201930,308388736639,1100569819672,3936511575349,14109420968044,50669532081790,182291946385519,656930486625718,2371133950246495
%N G.f. A(x) satisfies A(x) = A(x^3)/A(x^2) + A(x)^2.
%C Conjecture: a(n) == 1 (mod 3) for n >= 1, which has been verified for the initial 2305 terms.
%H Paul D. Hanna, <a href="/A377100/b377100.txt">Table of n, a(n) for n = 1..1200</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F (1) A(x) = A(x^3)/A(x^2) + A(x)^2.
%F (2) A(x) = 1 - A(x^3) / (A(x) * A(x^2)).
%F (3) A(x) = (1 - sqrt(1 - 4*A(x^3)/A(x^2)))/2.
%F a(n) ~ c * d^n / n^(3/2), where d = 3.7863653657737138633390523619934080121556430464612481222528... and c = 0.133726246724835700833882379294974176805097212863226248... - _Vaclav Kotesovec_, Nov 25 2024
%e G.f.: A(x) = x + x^2 + x^3 + 4*x^4 + 10*x^5 + 28*x^6 + 82*x^7 + 256*x^8 + 808*x^9 + 2617*x^10 + 8566*x^11 + 28447*x^12 + ...
%e where A(x) = A(x^3)/A(x^2) + A(x)^2.
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 29*x^6 + 84*x^7 + 256*x^8 + 812*x^9 + 2616*x^10 + 8578*x^11 + 28454*x^12 + ...
%e A(x^3)/A(x^2) = x - x^3 + x^4 - x^6 - 2*x^7 - 4*x^9 + x^10 - 12*x^11 - 7*x^12 - 20*x^13 - 12*x^14 - 106*x^15 - 11*x^16 - 282*x^17 - 133*x^18 - 863*x^19 - 318*x^20 - 3004*x^21 - 824*x^22 - 9642*x^23 - 3511*x^24 + ...
%e SPECIFIC VALUES.
%e A(t) = 2/5 at t = 0.2524344583306892446330618054730495682087506920909...
%e A(t) = 1/3 at t = 0.2320562211245459629892863344294626360139726683442...
%e A(t) = 1/4 at t = 0.1934098827751950108254656383626313546809677140215...
%e A(t) = 1/5 at t = 0.1636941944543552131282198725975208466904265162419...
%e A(1/4) = 0.38998631443325866799564283462093272794063010270972...
%e where A(1/4) = A(1/64)/A(1/16) + A(1/4)^2.
%e A(1/5) = 0.26231987380454237963569805198314775610358762307447...
%e where A(1/5) = A(1/125)/A(1/25) + A(1/5)^2.
%e A(1/6) = 0.20466907613859333014170015403398831627945711841689...
%e where A(1/6) = A(1/216)/A(1/36) + A(1/6)^2.
%e A(1/7) = 0.16886128781516765156942825490683640154708902665786...
%e A(1/8) = 0.14403120479636364458640343085117018181392306415283...
%e A(1/9) = 0.12568659456031966917982898016890665369785621094542...
%e A(1/16) = 0.06672301128588339314718493957102832603310943256142...
%e A(1/25) = 0.04167539404399936286416999003300893924018473834921...
%e A(1/36) = 0.02857337707390375424812158826059144303348530928252...
%e A(1/49) = 0.02083388767242775805806913860068416709771631936458...
%e A(1/64) = 0.01587320348112548054371664007880899788710197613384...
%e A(1/125) = 0.008064528719196404764792736015117367458367638921829...
%e A(1/216) = 0.004651164188289650656367523975284711447986222337517...
%o (PARI) \\ From formula: A(x) = A(x^3)/A(x^2) + A(x)^2 \\
%o {a(n) = my(A=[0,1]); for(m=1,n, A=concat(A,0);
%o A[#A] = polcoef( subst(Ser(A),x,x^3)/subst(Ser(A),x,x^2) - Ser(A) + Ser(A)^2, #A-1) );H=A;A[n+1]}
%o for(n=1,40,print1(a(n),", "))
%K nonn,new
%O 1,4
%A _Paul D. Hanna_, Nov 24 2024