Dear Math-Fun and Seqfans, In his 2002 survey, MR1898209 (2003c:34001) Ilyashenko, Yu. Centennial history of Hilbert's 16th problem. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 301--354 (electronic). (Reviewer: Lubomir Gavrilov) 34-02 (34C07 37C10 37F75) the author mentions that Petrovskii and Landis gave an incorrect proof that the Hilbert number H(n) is bounded above by P_3(n), "a certain polynomial of degree 3 [in n]". The reference is to: MR0073004 (17,364d) Petrovski\u\i, I. G.; Landis, E. M. On the number of limit cycles of the equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are polynomials of 2nd degree. (Russian) Mat. Sb. N.S. 37(79) (1955), 209--250. My question is, what is this sequence P_3(n) ? Later in the article Ilyashenko mentions a second Hilbert-type sequence E(n) (on page 305). Same question. NJAS