cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Jun 07 2014

Keywords

Comments

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.

Links

Crossrefs

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

Programs

Formula

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

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • Scheme
    (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)))

Formula

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

Views

Author

Antti Karttunen, Jan 27 2003

Keywords

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

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

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 07 2014

Keywords

Comments

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.

Links

Crossrefs

A243494 gives the same terms sorted into ascending order with duplicates removed.
Cf. A243496, A209636, A209637, A069787, A127301, A243490, A243491, A000079, A007097, A036256.

Formula

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).
Showing 1-4 of 4 results.

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