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 1-9 of 9 results.

A080976 a(n) = A080301(A080973(n)).

Original entry on oeis.org

0, 3, 318, 49497, 1593176614, 306347668831, 13663408341818193, 657273036441106021420, 1725515775006035094342654787041, 397951747550111200113555298627327
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Crossrefs

Cf. A080975.

A057505 Signature-permutation of a Catalan Automorphism: Donaghey's map M acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 6, 4, 22, 21, 18, 20, 17, 13, 12, 15, 19, 16, 10, 11, 14, 9, 64, 63, 59, 62, 58, 50, 49, 55, 61, 57, 46, 48, 54, 45, 36, 35, 32, 34, 31, 41, 40, 52, 60, 56, 43, 47, 53, 44, 27, 26, 29, 33, 30, 38, 39, 51, 42, 24, 25, 28, 37, 23, 196, 195, 190, 194, 189
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Comments

This is equivalent to map M given by Donaghey on page 81 of his paper "Automorphisms on ..." and also equivalent to the transformation procedure depicted in the picture (23) of Donaghey-Shapiro paper.
This can be also considered as a "more recursive" variant of A057501 or A057503 or A057161.

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

Inverse: A057506.
The 2nd, 3rd, 4th, 5th and 6th "power": A071661, A071663, A071665, A071667, A071669.
Other related permutations: A057501, A057503, A057161.
Cycle counts: A057507. Maximum cycle lengths: A057545. LCM's of all cycles: A060114. See A057501 for other Maple procedures.
Row 17 of table A122288.
Cf. A080981 (the "primitive elements" of this automorphism), A079438, A079440, A079442, A079444, A080967, A080968, A080972, A080272, A080292, A083929, A080973, A081164, A123050, A125977, A126312.

Programs

  • Maple
    map(CatalanRankGlobal,map(DonagheysM, A014486)); or map(CatalanRankGlobal,map(DeepRotateTriangularization, A014486));
DonagheysM := n -> pars2binexp(DonagheysMP(binexp2pars(n)));
DonagheysMP := h -> `if`((0 = nops(h)),h,[op(DonagheysMP(car(h))),DonagheysMP(cdr(h))]);
DeepRotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := DeepRotateTriangularization(BinTreeRightBranch(n))*2; z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;

Formula

a(0) = 0, and for n>=1, a(n) = A085201(a(A072771(n)), A057548(a(A072772(n)))). [This recurrence reflects the S-expression implementation given first in the Program section: A085201 is a 2-ary function corresponding to 'append', A072771 and A072772 correspond to 'car' and 'cdr' (known also as first/rest or head/tail in some languages), and A057548 corresponds to unary form of function 'list'].
As a composition of related permutations:
a(n) = A057164(A057163(n)).
a(n) = A057163(A057506(A057163(n))).

A080293 A014486-encoding of the branch-reduced binomial-mod-2 binary trees.

Original entry on oeis.org

2, 50, 14642, 3969842, 267572689202, 69427226972978, 4581045692538239282, 301220569271221714981682, 1295918094920364850246919050705202, 332029112115571675270693117549056818
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Comments

These are obtained from the stunted binomial-mod-2 zigzag trees (A080263) either by extending each leaf to a branch of two leaves, or by branch-reducing every other such tree.

Crossrefs

a(n) = A014486(A080295(n)). Same sequence in binary: A080294. Breadth-first-wise encoding: A080318. "Moose-trees" obtained from these: A080973. Cf. A080292, A080297.

Formula

a(n) = A080310(A080263(n)) = A014486(A080298(A080265(n))) = A014486(A080979(A080265(2*n)))

A080292 Orbit size of each tree A080293(n) under Donaghey's "Map M" Catalan automorphism.

Original entry on oeis.org

1, 3, 9, 9, 81, 81, 81, 27, 1701, 1701, 1701, 1701, 2673, 2673, 891, 891
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Comments

This is the size of the cycle containing A080295(n) in the permutations A057505/A057506.

Crossrefs

A080977(n) = A080272(2*n)/a(n). A080302(n) = a(n)/3 for n>0. Cf. A080973/A080975.

Formula

a(n) = A080967(A080295(n))

A080978 a(n) = 2*A006046(n) + 1.

Original entry on oeis.org

1, 3, 7, 11, 19, 23, 31, 39, 55, 59, 67, 75, 91, 99, 115, 131, 163, 167, 175, 183, 199, 207, 223, 239, 271, 279, 295, 311, 343, 359, 391, 423, 487, 491, 499, 507, 523, 531, 547, 563, 595, 603, 619, 635, 667, 683, 715, 747, 811, 819, 835, 851, 883, 899, 931, 963
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Comments

The number of edges in A080973-trees.
Conjectured partial sums of A131136. - Sean A. Irvine, Jun 25 2022

Links

Crossrefs

Cf. A006046, A080973.

Programs

A080975 A014486-index of the "Moose trees".

Original entry on oeis.org

1, 7, 515, 73211, 2249220471, 431283926958, 18838905762720934, 896134321804401371660, 2333852111980919691995847581921, 537961368577436933017494169487235
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Comments

See A080973 for illustration and comments.

Crossrefs

A057164(a(n)) = a(n) for all n. Cf. A080976.

Formula

a(n) = A057505^((A080292(n)+1)/2) (A080293(n)) [where ^ stands for the repeated applications of permutation A057505.]

A081155 Number of odd cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506, with fixed-points of both A057163 and A057164.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 3, 0, 6, 0, 12, 0, 20, 0, 59, 0, 120, 0
Offset: 0

Views

Author

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Keywords

Comments

See comment at A080973. Because A057163 can fix only binary trees of odd size, the even-indexed terms are all 0 after n>0.

Crossrefs

Bisection: A081156.
Cf. A081148, A081153.
Cf. A014137, A014138, A057505, A057506, A057163, A057164.

Formula

a(n) = A081148(n)-A081153(n).

A081156 Number of odd cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506, with fixed-points of both A057163 and A057164.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 20, 59, 120
Offset: 0

Views

Author

Wouter Meeussen & Antti Karttunen Mar 10 2003

Keywords

Comments

See comment at A080973.

Crossrefs

Cf. A080973.
Cf. A081155, A000108, A081163.
Cf. A014137, A014138, A057505, A057506, A057163, A057164.

Formula

a(n) = A081155(2n+1) = A000108(n) - 2*A081163(n).

A080974 A063171-encoding of the "Moose trees".

Original entry on oeis.org

10, 110100, 11101001101000, 1111010010011011010000, 11111010011010001001101110100110100000, 1111101001101101000010011011110100100110100000
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2003

Keywords

Formula

a(n) = A007088(A080973(n)).
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.