A243490 -id:A243490 - cp's OEIS frontend

A243493 Value of Matula-Goebel signature at the fixed points of A069787: a(n) = A127301(A243490(n)).

1, 2, 4, 3, 8, 6, 5, 16, 12, 10, 14, 13, 11, 32, 24, 20, 28, 26, 22, 37, 23, 34, 31, 64, 48, 40, 56, 52, 44, 74, 46, 68, 62, 76, 39, 89, 61, 47, 86, 101, 118, 109, 127, 128, 96, 80, 112, 104, 88, 148, 92, 136, 124, 152, 78, 178, 122, 94, 172, 202, 236, 218, 254

Offset: 0

Antti Karttunen, Jun 07 2014

The first duplicate value occurs at n=101, as a(101) = a(129) = 362. The corresponding A014486-indices are A243490(101) = 924 and A243490(129) = 1640, respectively.

Antti Karttunen, Table of n, a(n) for n = 0..525

A243494 gives the same terms sorted into ascending order with duplicates removed.

Cf. A243496, A209636, A209637, A069787, A127301, A243490, A243491, A000079, A007097, A036256.

a(n) = A127301(A243490(n)).
a(n) = A243491(A243490(n)).
For all n >= 0, a(A036256(n)-1) = A007097(n) and a(A036256(n)) = A000079(n+1).

A243489 Complement of A243490: Numbers n such that A069787(n) <> n.

Original entry on oeis.org

6, 7, 11, 12, 14, 15, 17, 18, 19, 21, 25, 26, 28, 29, 31, 32, 33, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 63, 67, 68, 70, 71, 73, 74, 75, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 99, 100

Offset: 1

Antti Karttunen, Jun 07 2014

nonn

Numbers that are not fixed by permutation A069787.

Antti Karttunen, Table of n, a(n) for n = 1..6644

Complement: A243490. Cf. A069787.

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

Original entry on oeis.org

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

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).

A243491 Matula-Goebel signature computed for trees rearranged by Catalan automorphism *A069787: a(n) = A127301(A069787(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 7, 6, 5, 16, 12, 14, 12, 10, 19, 13, 14, 12, 9, 17, 13, 10, 11, 32, 24, 28, 24, 20, 38, 26, 28, 24, 18, 34, 26, 20, 22, 53, 37, 43, 37, 29, 38, 26, 28, 24, 18, 21, 21, 18, 15, 67, 41, 43, 37, 23, 34, 26, 20, 15, 59, 41, 29, 22, 31, 64, 48, 56, 48, 40

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

See the comments at A243492.

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

A243492 gives the differences from A127301. Cf. also A243490, A243493.

(define (A243491 n) (A127301 (A069787 n)))

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

cp's OEIS Frontend

A243493 Value of Matula-Goebel signature at the fixed points of A069787: a(n) = A127301(A243490(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A243489 Complement of A243490: Numbers n such that A069787(n) <> n.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A243491 Matula-Goebel signature computed for trees rearranged by Catalan automorphism *A069787: a(n) = A127301(A069787(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula