

A079438


a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n10)/4) + floor((n14)/8))).


9



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

0,3


COMMENTS

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(n1)..A014138(n1)] such that A057164(i) = i and A057164(A057163(i)) = A057163(i).
(Thus also) the number of fixed points in range [A014137(n1)..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.
Additional comment from Antti Karttunen, Dec 13 2017: (Start)
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)


REFERENCES

D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All TreesHistory of Combinatorial Generation, vi+120pp. ISBN 0321335708 AddisonWesley Professional; 1ST edition (Feb 06, 2006).


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000
R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 7590.
A. Karttunen, Cprogram for counting the initial terms of this sequence (empirically)
A. Karttunen, Illustration of initial terms for trees of sizes n=2..18
A. Karttunen, On the fixed points of A071661 (Notes in OEIS Wiki)
D. E. Knuth, PreFascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6.


FORMULA

a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n10)/4) + floor((n14)/8))).


MAPLE

A079438 := n > `if`((n<2), 1, 2*(floor((n+1)/3) + `if`((n>=14), floor((n10)/4)+floor((n14)/8), 0)));


MATHEMATICA

a[0]:= 1; a[1]:= 1; a[n_]:= a[n] = 2*Floor[(n+1)/3] +2*If[ n >= 14, (Floor[(n10)/4] +Floor[(n14)/8]), 0]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, Jan 18 2019 *)


PROG

(PARI) {a(n) = if(n==0, 1, if(n==1, 1, 2*floor((n+1)/3) + 2*if(n >= 14, floor( (n10)/4) + floor((n14)/8), 0)))}; \\ G. C. Greubel, Jan 18 2019


CROSSREFS

Cf. A000108, A057163, A057164, A057505, A069787, A071661, A079437, A079439, A079442, A080070, A243490, A243491, A243492.
From n>= 2 onward A079440(n) = a(n)/2.
Occurs in A073202 as row 13373289.
Differs from A123050 for the first time at n=26.
Sequence in context: A323094 A086227 A302402 * A123050 A113694 A086159
Adjacent sequences: A079435 A079436 A079437 * A079439 A079440 A079441


KEYWORD

nonn


AUTHOR

Antti Karttunen, Jan 27 2003


EXTENSIONS

Entry edited (the definition replaced by a formula, the old definition moved to the comments)  Antti Karttunen, Dec 13 2017


STATUS

approved



