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 A344623 Pseudo-involution companion for the Fibonacci generating function. 1
 1, 3, 9, 32, 126, 538, 2429, 11412, 55201, 272993, 1373784, 7011297, 36201841, 188761743, 992491049, 5256244537, 28013213196, 150128293038, 808543940999, 4373798584407, 23753913152691, 129469596050953, 707969244301884, 3882857013894482, 21353585584100401 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is the number of colored Schröder paths of semilength n with steps U=(1,0) and D=(1,-1) of 1 color and H=(2,0) of 2 colors, red and blue, where H does not follow D, and no two red H steps are consecutive. LINKS Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021. FORMULA G.f.: A(x) satisfies A(-x*A(x)) = 1/A(x) and F(-x*A(x)) = 1/F(x), where x*F(x) is g.f. of A000045. I.e., the Riordan array (F(x), x*A(x)) is a pseudo-involution. G.f.: A(x) = (F(x) - 1)*C(F(x) - 1)/x, where C(x) is the g.f. of A000108 and x*F(x) is g.f. of A000045. G.f.: A(x) = (1 - sqrt((1 - 5*x - 5*x^2)/(1 - x - x^2)))/(2x). G.f.: Let B(x) = 2 + g.f.(A200031(n)), then A(x) = 1 + x*A(x)*B(x^2*A(x)). a(n) ~ sqrt(3) * 5^(n/2 + 1) * phi^(2*n + 1) / (8*sqrt(Pi)*n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 25 2021 CROSSREFS Cf. A000045, A000108, A200031. Sequence in context: A104184 A339230 A193621 * A320180 A183425 A039628 Adjacent sequences: A344620 A344621 A344622 * A344624 A344625 A344626 KEYWORD nonn AUTHOR Alexander Burstein, May 24 2021 STATUS approved

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Last modified February 2 05:15 EST 2023. Contains 359997 sequences. (Running on oeis4.)