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A079438
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a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n-10)/4) + floor((n-14)/8))).
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10
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1, 1, 2, 2, 2, 4, 4, 4, 6, 6, 6, 8, 8, 8, 12, 12, 12, 14, 16, 16, 18, 18, 22, 24, 24, 24, 28, 28, 28, 30, 34, 34, 36, 36, 38, 40, 40, 40, 46, 46, 46, 48, 50, 50, 52, 52, 56, 58, 58, 58, 62, 62, 62, 64, 68, 68, 70, 70, 72, 74, 74, 74, 80, 80, 80, 82, 84, 84, 86, 86, 90, 92, 92, 92
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OFFSET
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0,3
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COMMENTS
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The original definition was: Number of rooted general plane trees which are symmetric and will stay symmetric after the underlying plane binary tree has been reflected, i.e., number of integers i in range [A014137(n-1)..A014138(n-1)] such that A057164(i) = i and A057164(A057163(i)) = A057163(i).
(Thus also) the number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071661 (= Donaghey's automorphism M "squared"), which is equal to condition A057164(i) = A069787(i) = i, i.e., the size of the intersection of fixed points of permutations A057164 and A069787 in the same range.
However, David Callan's A123050 claims to give more correct version of that count from n=26 onward, so I probably made a little mistake when converting my insights into the formula given here. At that time I reckoned that if the conjecture given in A080070 were true, then it would imply that the formula given here were exact, otherwise it would give only a lower bound.
It would be nice to know what an empirical program would give as the count of fixed points of A071661 for n in range [A014137(25)..A014138(26)] = [6619846420553 .. 24987199492704], with total A000108(26) = 18367353072151 points to check.
(End)
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees--History of Combinatorial Generation, vi+120pp. ISBN 0-321-33570-8 Addison-Wesley Professional; 1ST edition (Feb 06, 2006).
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LINKS
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FORMULA
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a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n-10)/4) + floor((n-14)/8))).
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MAPLE
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A079438 := n -> `if`((n<2), 1, 2*(floor((n+1)/3) + `if`((n>=14), floor((n-10)/4)+floor((n-14)/8), 0)));
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MATHEMATICA
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a[0]:= 1; a[1]:= 1; a[n_]:= a[n] = 2*Floor[(n+1)/3] +2*If[ n >= 14, (Floor[(n-10)/4] +Floor[(n-14)/8]), 0]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, Jan 18 2019 *)
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PROG
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(PARI) {a(n) = if(n==0, 1, if(n==1, 1, 2*floor((n+1)/3) + 2*if(n >= 14, floor( (n-10)/4) + floor((n-14)/8), 0)))}; \\ G. C. Greubel, Jan 18 2019
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CROSSREFS
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Cf. A000108, A057163, A057164, A057505, A069787, A071661, A079437, A079439, A079442, A080070, A243490, A243491, A243492.
From n>= 2 onward A079440(n) = a(n)/2.
Differs from A123050 for the first time at n=26.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Entry edited (the definition replaced by a formula, the old definition moved to the comments) - Antti Karttunen, Dec 13 2017
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STATUS
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approved
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