A079438 a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n-10)/4) + floor((n-14)/8))).

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

Offset: 0

Author

Antti Karttunen, Jan 27 2003

Keywords

• nonn

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(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.

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 Trees--History of Combinatorial Generation, vi+120pp. ISBN 0-321-33570-8 Addison-Wesley 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), 75-90.
• A. Karttunen, C-program 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, Pre-Fascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6.

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.

Programs

Maple

A079438 := n -> `if`((n<2),1,2*(floor((n+1)/3) + `if`((n>=14),floor((n-10)/4)+floor((n-14)/8),0)));

Mathematica

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 *)

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

Formula

a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n-10)/4) + floor((n-14)/8))).

Extensions

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