A243492 -id:A243492 - cp's OEIS frontend

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

			A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to Matula-Goebel numbers

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(|A106191(n)|) = A033844(n-1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
For standard instead of binary encoding we have A358506.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists binary encodings of ordered rooted trees.
Cf. A001263, A061775, A196050, A206487, A358505.

mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)

(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

A001222(a(n)) = A057515(n) for all n.

A324055 Deficiency of Doudna-sequence: a(n) = A033879(A005940(1+n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 5, 1, 6, 2, 6, -4, 19, -3, 14, 1, 10, 4, 10, -2, 22, -12, 12, -12, 41, 7, 26, -19, 94, -12, 41, 1, 12, 8, 18, 0, 38, -12, 22, -10, 58, -4, 18, -48, 102, -54, 30, -28, 109, 25, 66, -17, 148, -72, 47, -51, 286, 32, 126, -64, 469, -39, 122, 1, 16, 10, 22, 4, 46, -12, 42, -8, 70, 4, 42, -56, 178, -60, 58, -26, 118, 20

Offset: 0

Antti Karttunen, Feb 14 2019

Both here and in the mirror image sequence A324185, the lowermost (asinh) scatter plot shows on the y = 0 line the numbers that correspond to the perfect numbers. Compare also to the scatter plot of A243492.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A000120, A033879, A054429.
See A106737, A290077, A323915, A324052, A324054, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349, A324394, A324395 for other sequences as permuted by A005940, and compare their scatter plots.
Cf. also A243492, A323174, A323244, A324185, A324346, A324347, A324654.

Array[Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2]}, 2 # - DivisorSigma[1, #] &[Times @@ Flatten@ Table[Prime[Count[Flatten@ #, 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]] &, 82, 0] (* Michael De Vlieger, Mar 11 2019, after Robert G. Wilson v at A005940 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A033879(n) = (2*n-sigma(n));
A324055(n) = A033879(A005940(1+n));

A324055(n) = { my(m1=2,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); (m1-m2); };

a(n) = A033879(A005940(1+n)).
a(n) = 2*A005940(1+n) - A324054(n).
For n > 0, a(n) = A324185(A054429(n)).
a(n) = A324348(n) + A000120(A005940(1+n)).

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

A243490 Fixed points of A069787: Numbers n such that A069787(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 10, 13, 16, 20, 22, 23, 24, 27, 30, 34, 36, 54, 55, 56, 64, 65, 66, 69, 72, 76, 78, 96, 97, 98, 106, 126, 136, 157, 158, 162, 165, 183, 186, 193, 196, 197, 198, 201, 204, 208, 210, 228, 229, 230, 238, 258, 268, 289, 290, 294, 297, 315

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

Although in principle a list, the indexing of this sequence starts from zero, as 0 is always fixed by all Catalan bijections (permutations induced by bijective operations performed on A014486), so it is a trivial case, which can be skipped by considering only values from a(n>=1) onward.

Sequence gives also the positions of all zeros in A243492.

Antti Karttunen, Table of n, a(n) for n = 0..7059

Complement: A243489.
Fixed points of A069787, positions of zeros in A243492.
Cf. also A001405, A036256, A061856, A126312, A079442, A243493.

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

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

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A324055 Deficiency of Doudna-sequence: a(n) = A033879(A005940(1+n)).

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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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A243490 Fixed points of A069787: Numbers n such that A069787(n) = n.

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

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