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A065164
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Permutation t->t+1 of Z, folded to N.
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6
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2, 4, 1, 6, 3, 8, 5, 10, 7, 12, 9, 14, 11, 16, 13, 18, 15, 20, 17, 22, 19, 24, 21, 26, 23, 28, 25, 30, 27, 32, 29, 34, 31, 36, 33, 38, 35, 40, 37, 42, 39, 44, 41, 46, 43, 48, 45, 50, 47, 52, 49, 54, 51, 56, 53, 58, 55, 60, 57, 62, 59, 64, 61, 66, 63, 68, 65, 70, 67, 72, 69, 74
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OFFSET
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1,1
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COMMENTS
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Corresponds to simple periodic asynchronic site swap pattern ...111111... (tossing one ball from hand to hand forever).
This permutation consists of a single infinite cycle.
This is, starting at a(2) = 4, the same as the "increasing oscillating sequence" shown in Proposition 3.1, p.7 and plotted in the right of figure 1, of Vatter. The same paper, p.4, cites Comtet and uses without giving the A-number of A003319. Abstract: We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley-Wilf limit) at least lambda = approx 2.48187, the unique real root of x^5-2x^4-2x^2-2x-1, thereby establishing a conjecture of Albert and Linton. - Jonathan Vos Post, Jul 18 2008
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 819.
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LINKS
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FORMULA
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Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)+1).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
G.f.: x*(3*x^3-5*x^2+2*x+2) / ((x-1)^2*(x+1)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 1. - Amiram Eldar, Aug 08 2023
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MAPLE
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ss1 := [seq(PerSS(n, 1), n=1..120)]; PerSS := (n, c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1), 1, 0);
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MATHEMATICA
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Join[{2}, LinearRecurrence[{1, 1, -1}, {4, 1, 6}, 100]] (* Amiram Eldar, Aug 08 2023 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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