A057548 -id:A057548 - cp's OEIS frontend

A057547 A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

2, 12, 52, 56, 212, 216, 228, 232, 240, 852, 856, 868, 872, 880, 916, 920, 932, 936, 944, 964, 968, 976, 992, 3412, 3416, 3428, 3432, 3440, 3476, 3480, 3492, 3496, 3504, 3524, 3528, 3536, 3552, 3668, 3672, 3684, 3688, 3696, 3732, 3736, 3748, 3752, 3760

Offset: 0

Antti Karttunen Sep 07 2000

This one-to-one correspondence between all rooted plane trees and one node larger, root degree = 1 trees illustrates the fact that INVERT(A000108) = LEFT(A000108). (Catalan numbers shift left under Cameron's A transformation.)
From Ruud H.G. van Tol, May 13 2024: (Start)
Sequence on a lattice:
Tree         Paths                  Decimal  Count
|_           10                           2      1
|.         1100                        12      1
||._       110100    -111000        52,56      2
|||_._     11010100  -11110000    212-240      5
|||_|.   1101010100-1111100000  852-992     14
... (End)

Ruud H.G. van Tol, Table of n, a(n) for n = 0..2055
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Index entries for encodings of plane rooted trees

Double-trunked trees: A057517. Cf. also A057548, A057549.

alltrees2singletrunked := n -> pars2binexp([binexp2pars(n)]); # Just surround with extra parentheses.

a_rows(N) = my(a=Vec([[2]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, for(i=1, valuation(t, 2), listput(~c, (t<<2)+(2<Ruud H.G. van Tol, May 25 2024

a(n) = A014486(A057548(n)) and also from n > 0 onward = A079946(A014486(n)).

a(n) = alltrees2singletrunked(A014486[n]) (see Maple code below and in A057501).

A075164 Position of A014486(n-1) in A075165.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 16, 7, 10, 15, 12, 24, 25, 18, 27, 32, 64, 81, 512, 256, 65536, 11, 14, 21, 20, 40, 35, 30, 45, 48, 96, 135, 768, 384, 98304, 49, 50, 75, 36, 72, 125, 54, 243, 128, 1024, 729, 32768, 4096, 16777216, 625, 162, 19683, 33554432, 262144

Offset: 1

Antti Karttunen, Sep 13 2002

nonn

See A075166.

Index entries for sequences that are permutations of the natural numbers

Inverse of A075163. a(n) = A075162(n-1)+1.

a(n) = A106443(A106454(n)). A000040(n) = a(1+A014137(n)). The powers of two are located at indices given by A057548 + 1, permuted by this same sequence, i.e. a(A057548(n)+1) = A000079(a(n)). - Antti Karttunen, May 09 2005

A085205 Array A(x,y): induced by the 2-ary form of the list-function 'list' present in the programming languages Lisp and Scheme, in the same way as A085201 is induced by the 2-ary 'append'-function. Listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

Original entry on oeis.org

2, 6, 5, 16, 15, 12, 19, 43, 40, 13, 44, 52, 124, 41, 31, 47, 127, 152, 125, 115, 32, 53, 136, 388, 153, 379, 116, 34, 56, 155, 416, 389, 469, 380, 118, 35, 60, 164, 478, 417, 1237, 470, 382, 119, 36, 128, 178, 506, 479, 1327, 1238, 472, 383, 120, 87, 131, 391

Offset: 0

Antti Karttunen, Jun 23 2003

Index entries for the sequences induced by list functions of Lisp

Transpose: A085206. Row 0: A085226, Column 0: A085227, Diagonal: A085228.

a(x, y) = A072764bi(x, A057548(y)).

A376402 Bitwise XOR (centrally aligned) of two consecutive terms of A122242.

Original entry on oeis.org

164, 628, 2444, 10040, 34424, 142400, 612536, 2536016, 8772720, 36320296, 156298040, 648930320, 2246427920, 9290072680, 40123676576, 166398412640, 574717970240, 2376856817864, 10244120543704, 42544644116352, 146496800436256, 607708669110320, 2625008220416552, 10882360875506928, 37586414897168848, 156056124134144296

Offset: 1

Antti Karttunen, Sep 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(2048) drawn as binary strings, in Wolframesque fashion.
Index entries for sequences related to binary expansion of n

Cf. A122242 (A014486, A057548, A082358, A122237, A122241).

Cf. also A327973, A376405, A376412.

A376402(n) = bitxor(A122242(1+n), 2*A122242(n));

Python
```
# See the Links section
```

a(n) = A122242(1+n) XOR 2*A122242(n).

A376412 a(n) = A122242(4+n) XOR 16*A122242(n).

Original entry on oeis.org

14544, 64920, 258096, 925720, 3703264, 16562224, 66158664, 237396008, 948568384, 4239130768, 16949182920, 60767078320, 243080890624, 1085016114240, 4341071150792, 15535530051144, 62225888982288, 277421534227968, 1111070191401136, 3979658311943880, 15908408006551904, 71162952082313488, 284082756324759560, 1019946695587234480

Offset: 1

Antti Karttunen, Sep 22 2024

See comments in A376415.

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(768) drawn as binary strings, in Wolframesque fashion.
Antti Karttunen, The central parts of terms a(1) - a(20000) drawn in similar fashion. (the widening blank corridor in the middle corresponds with the "central crystallization" attractor present in A122242)
Index entries for sequences related to binary expansion of n

Cf. A122242 (A014486, A057548, A082358, A122237, A122241).

Cf. also A327973, A376402, A376415.

A376412(n) = bitxor(A122242(4+n), 16*A122242(n));

Python
```
# See the Links section
```

a(n) = A122242(4+n) XOR 16*A122242(n).

A057549 The local ranks of each term of A057547.

Original entry on oeis.org

0, 1, 3, 4, 8, 9, 11, 12, 13, 22, 23, 25, 26, 27, 31, 32, 34, 35, 36, 38, 39, 40, 41, 64, 65, 67, 68, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 92, 93, 95, 96, 97, 101, 102, 104, 105, 106, 108, 109, 110, 111, 115, 116, 118, 119, 120, 122, 123, 124, 125, 127, 128, 129

Offset: 0

Antti Karttunen, Sep 07 2000

nonn

Cf. A057519, A057548, A057550.

a(n) = CatalanRank(floor(binwidth(A057547[n])/2), A057547[n])

A376405 Bitwise XOR (centrally aligned) of two consecutive terms of A122245.

Original entry on oeis.org

176, 584, 2068, 9232, 38952, 135296, 567096, 2444568, 10136288, 34920688, 144732808, 625792608, 2598216176, 8989149792, 37119010736, 160422522664, 665656629200, 2297815400576, 9505629937992, 41066855413976, 169932530966160, 589165636912400, 2439104800321640, 10514745879265952, 43543845360254320, 149771860125187648

Offset: 1

Antti Karttunen, Sep 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(2048) drawn as binary strings, in Wolframesque fashion.
Index entries for sequences related to binary expansion of n

Cf. A122245 (A014486, A057548, A082358, A122237, A122244).

Cf. also A327973, A376402, A376415.

A376405(n) = bitxor(A122245(1+n), 2*A122245(n));

Python
```
# See the Links section
```

a(n) = A122245(1+n) XOR 2*A122245(n).

A376415 a(n) = A122245(4+n) XOR 16*A122245(n).

Original entry on oeis.org

14488, 57880, 258096, 1033752, 3702824, 14821424, 66158736, 264831016, 948570032, 3798139024, 16949183552, 67829237680, 243080889928, 972279514176, 4341071097344, 17360471721544, 62225889019592, 248568875068928, 1111070190653712, 4438793067349704, 15908408008868528, 63634253845942544, 284082756299099488, 1136310075423425200

Offset: 1

Antti Karttunen, Sep 22 2024

This seems to preserve more of the "wavy texture" present in A122245 than what A376412 does vis-a-vis A122242. Compare the corresponding illustrations.

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(768) drawn as binary strings, in Wolframesque fashion.
Antti Karttunen, The central parts of terms a(1) - a(20000) drawn in similar fashion. (the widening blank region in the middle corresponds with the "central crystallization" attractor present in A122245)
Index entries for sequences related to binary expansion of n

Cf. A122245 (A014486, A057548, A082358, A122237, A122244).

Cf. also A327973, A376405, A376412.

A376415(n) = bitxor(A122245(4+n), 16*A122245(n));

Python
```
# See the Links section
```

A083930 Map from binary trees of size n to the set of corresponding trivalent plane trees (tpt) represented as size 2n+1 general trees.

Original entry on oeis.org

0, 7, 49, 57, 439, 452, 515, 541, 585, 4612, 4631, 4757, 4795, 4865, 5455, 5468, 5744, 5795, 5865, 6268, 6294, 6433, 6688, 53200, 53226, 53448, 53500, 53604, 54935, 54954, 55430, 55501, 55605, 56356, 56394, 56601, 57003, 63294, 63313, 63439

Offset: 0

Antti Karttunen, May 13 2003

nonn

Inverse function: A083929. Positions of A083936 in A014486.

a(n) = A057548(A057123(n)).

A153242 Positions of general trees in A014486 whose degree is not one.

Original entry on oeis.org

0, 2, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 19, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 51, 52, 53, 56, 60, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93

Offset: 0

Antti Karttunen, Dec 21 2008

nonn

A. Karttunen, Table of n, a(n) for n = 0..1000

I.e. such i that A057515(i) is not 1. Complement of A057548. For n>0, gives the indices of trees whose degree is greater than one. (At least two top-level branches).

cp's OEIS Frontend

A057547 A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

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Maple

PARI

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A075164 Position of A014486(n-1) in A075165.

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A085205 Array A(x,y): induced by the 2-ary form of the list-function 'list' present in the programming languages Lisp and Scheme, in the same way as A085201 is induced by the 2-ary 'append'-function. Listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

Views

Author

Keywords

Links

Crossrefs

Formula

A376402 Bitwise XOR (centrally aligned) of two consecutive terms of A122242.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A376412 a(n) = A122242(4+n) XOR 16*A122242(n).

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

A057549 The local ranks of each term of A057547.

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Keywords

Crossrefs

Formula

A376405 Bitwise XOR (centrally aligned) of two consecutive terms of A122245.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A376415 a(n) = A122245(4+n) XOR 16*A122245(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

A083930 Map from binary trees of size n to the set of corresponding trivalent plane trees (tpt) represented as size 2n+1 general trees.

Views

Author

Keywords

Crossrefs

Formula