A080070 -id:A080070 - cp's OEIS frontend

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

Antti Karttunen, Jan 27 2003

nonn

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)

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).

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.

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.

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

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 *)

{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

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

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

A080068 Iterates of A080067.

Original entry on oeis.org

0, 1, 2, 5, 11, 34, 82, 287, 923, 3016, 8664, 37407, 102983, 414050, 1488140, 5330976, 19367263, 76887675, 255935344, 808348061, 3420607749, 14063447808, 52376853939, 178612888734, 666146253450, 2182739853536, 10254827577242, 31021294603020, 132781022354213

Offset: 0

Antti Karttunen, Jan 27 2003

Corresponding totally balanced binary sequences in A063171: A080070.

Cf. A080067.

a(0) = 0, a(n) = A080067(a(n-1)).

More terms from Sean A. Irvine, Sep 02 2025

A122230 a(n) = A007088(A122229(n)).

Original entry on oeis.org

0, 10, 1100, 111000, 11100100, 1110011000, 111001100100, 11100110011000, 1110011001100100, 111001100110011000, 11100110011001100100, 1110011001100110011000, 111001100110011001100100

Offset: 0

Antti Karttunen, Sep 14 2006

Cf. A080070, A122233, A122236, A122240, A122243, A122246.

A122233 a(n) = A007088(A122232(n)).

Original entry on oeis.org

101010, 11010100, 1111100000, 111100100100, 11110001011000, 1111110010000100, 111101100010011000, 11110001101100100100, 1111101011000110000100, 111101100110001100010100

Offset: 1

Antti Karttunen, Sep 14 2006

Cf. A080070, A122230, A122236, A122240, A122243, A122246.

A122243 a(n) = A007088(A122242(n)).

Original entry on oeis.org

101010, 11110000, 1110010100, 111010100100, 11101001110000, 1111001010011000, 111100100101110000, 11101100101001011000, 1110110010011011100000, 111100111001000110110000

Offset: 1

Antti Karttunen, Sep 14 2006

A080070, A122230, A122233, A122236, A122240, A122246.

A122246 a(n) = A007088(A122245(n)).

Original entry on oeis.org

101100, 11101000, 1110011000, 111100100100, 11101001011000, 1110110010011000, 111100100110110000, 11110011010001011000, 1110110010010110101000, 111011001110000110110000

Offset: 1

Antti Karttunen, Sep 14 2006

A080070, A122230, A122233, A122236, A122240, A122243.

A122236 a(n) = A007088(A122235(n)).

Original entry on oeis.org

101100, 11011000, 1111001000, 111100010100, 11111100000100, 1111001001110000, 111100010111000100, 11111111000010000100, 1111100010011011000100, 111101101001110000100100

Offset: 1

Antti Karttunen, Sep 14 2006

Cf. A080070, A122230, A122233, A122240, A122243, A122246.

A122240 a(n) = A007088(A122239(n)).

Original entry on oeis.org

110100, 11110000, 1111000100, 111110000100, 11110011000100, 1111110000010100, 111100100111100000, 11110001011100100100, 1111111000101100000100, 111100101001101001110000

Offset: 1

Antti Karttunen, Sep 14 2006

Cf. A080070, A122230, A122233, A122236, A122243, A122246.

A243492 Difference A243491(n) - A127301(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 2, -2, 0, 7, 4, 0, -7, -4, 7, 0, -7, 0, 0, 0, 4, -4, 0, 14, 8, 0, -14, -8, 14, 0, -14, 0, 29, 19, 25, 16, 14, 10, 5, -10, -29, -19, -5, -16, -25, -14, 47, 26, 17, 0, 0, 0, -17, -47, -26, 37, 12, -12, -37, 0, 0, 0, 8, -8, 0, 28, 16, 0, -28, -16, 28, 0, -28, 0

Offset: 0

Antti Karttunen, Jun 07 2014

A243490 gives the positions of zeros, which are also the fixed points of A069787. They correspond to the dots shown on the y=0 line of the arcsinh-version of scatter plot.

Antti Karttunen, Table of n, a(n) for n = 0..6917
A. Karttunen, On the fixed points of A071661 (Notes in OEIS Wiki)

Cf. A069787, A243490, A057505, A079438, A123050, A080070, A153239, A153240, A153241, A154477.

(define (A243492 n) (- (A243491 n) (A127301 n)))

a(n) = A243491(n) - A127301(n) = A127301(A069787(n)) - A127301(n).

A080067 a(n) = A057163(A057548(A057164(n))).

Original entry on oeis.org

1, 2, 5, 4, 13, 11, 12, 10, 9, 36, 33, 34, 29, 28, 35, 30, 32, 27, 25, 31, 26, 24, 23, 106, 102, 103, 94, 93, 104, 95, 97, 83, 81, 96, 82, 80, 79, 105, 98, 99, 85, 84, 101, 89, 92, 78, 75, 90, 76, 71, 70, 100, 86, 91, 77, 72, 88, 74, 69, 67, 87, 73, 68, 66, 65, 328, 323, 324

Offset: 0

Antti Karttunen, Jan 27 2003

nonn

Iterates starting from zero: A080068. Cf. A080070.

cp's OEIS Frontend

A079438 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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A080068 Iterates of A080067.

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A122230 a(n) = A007088(A122229(n)).

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A122233 a(n) = A007088(A122232(n)).

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A122243 a(n) = A007088(A122242(n)).

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A122246 a(n) = A007088(A122245(n)).

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A122236 a(n) = A007088(A122235(n)).

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A122240 a(n) = A007088(A122239(n)).

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A243492 Difference A243491(n) - A127301(n).

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A080067 a(n) = A057163(A057548(A057164(n))).

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