A377252 - OEIS

%I #12 Nov 19 2024 00:51:42

%S 1,1,1,2,4,7,12,22,42,81,157,307,606,1206,2416,4865,9839,19981,40737,

%T 83343,171028,351940,726099,1501642,3112400,6464125,13450825,28038767,

%U 58544953,122431896,256408712,537732762,1129175346,2374028444,4997020292,10529562040,22210529816,46895830078,99109479009

%N G.f. satisfies A(x) = A( x^3 + x^3*A(x) ) / x^2.

%H Paul D. Hanna, <a href="/A377252/b377252.txt">Table of n, a(n) for n = 1..730</a>

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

%F (1) A(x) = A( x^3*(1 + A(x)) ) / x^2.

%F (2) A(x) = A( x^9*(1 + A(x))^3*(1 + x^2*A(x)) ) / ( x^8*(1 + A(x))^2 ).

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 12*x^7 + 22*x^8 + 42*x^9 + 81*x^10 + 157*x^11 + 307*x^12 + 606*x^13 + 1206*x^14 + ...

%e where A( x^3*(1 + A(x)) ) = x^2*A(x).

%e Also, A( x^9*(1 + A(x))^3*(1 + x^2*A(x)) ) = x^8*(1 + A(x))^2*A(x) = x^9 + 3*x^10 + 6*x^11 + 11*x^12 + 22*x^13 + 46*x^14 + 94*x^15 + 188*x^16 + ...

%e SPECIFIC VALUES.

%e A(t) = 3/2 at t = 0.45192181062521801557126190643383029659754110164264...

%e A(t) = 1   at t = 0.42253723660811532721783592250513176051852940714141...

%e   where A(2*t^3) = t^2.

%e A(t) = 4/5 at t = 0.39461010356746001740692189277728352723457637457033...

%e A(t) = 3/4 at t = 0.38522091778551420561548609384297493798523043797274...

%e A(t) = 2/3 at t = 0.36684501490591205836279362547950880677986553538331...

%e A(t) = 1/2 at t = 0.31731913114014607672084815330848095240076504967806...

%e A(t) = 1/3 at t = 0.24509058978713537427030162951676759822656915027917...

%e A(t) = 1/4 at t = 0.19805752511932025709405041056127061135572689011700...

%e A(t) = 1/5 at t = 0.16575581149357914101340688777414281331287359812793...

%e A(t) = -1/4 at t = -0.34347257074458628349336868043177584333373207788086...

%e A(t) = -1/5 at t = -0.25324896794350086294861429681341330954101228640025...

%e A(2/5) = 0.83169386298208509165369971723990921631081372563860...

%e A(1/3) = 0.54719854999222390328279014536372378999431212769540...

%e A(3/10) = 0.45407977578501996360299871474219423309393650280054...

%e A(1/4) = 0.34293211823255431017796847098989297339303439821881...

%e A(1/5) = 0.25317301806304124202078499350762927150976225191682...

%o (PARI) {a(n) = my(A=[0,1], Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);

%o A[#A] = polcoef( subst(Ax,x, x^3 + x^3*Ax) - x^2*Ax, #A+1) ); A[n+1]}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A091600.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Nov 18 2024