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A377252
G.f. satisfies A(x) = A( x^3 + x^3*A(x) ) / x^2.
1
1, 1, 1, 2, 4, 7, 12, 22, 42, 81, 157, 307, 606, 1206, 2416, 4865, 9839, 19981, 40737, 83343, 171028, 351940, 726099, 1501642, 3112400, 6464125, 13450825, 28038767, 58544953, 122431896, 256408712, 537732762, 1129175346, 2374028444, 4997020292, 10529562040, 22210529816, 46895830078, 99109479009
OFFSET
1,4
LINKS
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = A( x^3*(1 + A(x)) ) / x^2.
(2) A(x) = A( x^9*(1 + A(x))^3*(1 + x^2*A(x)) ) / ( x^8*(1 + A(x))^2 ).
EXAMPLE
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 + ...
where A( x^3*(1 + A(x)) ) = x^2*A(x).
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 + ...
SPECIFIC VALUES.
A(t) = 3/2 at t = 0.45192181062521801557126190643383029659754110164264...
A(t) = 1 at t = 0.42253723660811532721783592250513176051852940714141...
where A(2*t^3) = t^2.
A(t) = 4/5 at t = 0.39461010356746001740692189277728352723457637457033...
A(t) = 3/4 at t = 0.38522091778551420561548609384297493798523043797274...
A(t) = 2/3 at t = 0.36684501490591205836279362547950880677986553538331...
A(t) = 1/2 at t = 0.31731913114014607672084815330848095240076504967806...
A(t) = 1/3 at t = 0.24509058978713537427030162951676759822656915027917...
A(t) = 1/4 at t = 0.19805752511932025709405041056127061135572689011700...
A(t) = 1/5 at t = 0.16575581149357914101340688777414281331287359812793...
A(t) = -1/4 at t = -0.34347257074458628349336868043177584333373207788086...
A(t) = -1/5 at t = -0.25324896794350086294861429681341330954101228640025...
A(2/5) = 0.83169386298208509165369971723990921631081372563860...
A(1/3) = 0.54719854999222390328279014536372378999431212769540...
A(3/10) = 0.45407977578501996360299871474219423309393650280054...
A(1/4) = 0.34293211823255431017796847098989297339303439821881...
A(1/5) = 0.25317301806304124202078499350762927150976225191682...
PROG
(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = polcoef( subst(Ax, x, x^3 + x^3*Ax) - x^2*Ax, #A+1) ); A[n+1]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
Cf. A091600.
Sequence in context: A190165 A127542 A023432 * A072641 A280352 A135360
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 18 2024
STATUS
approved