A218778 -id:A218778 - cp's OEIS frontend

A218614 a(n) = binary code (shown here in decimal) of the position of natural number n in the beanstalk-tree A218778.

1, 2, 3, 5, 7, 9, 13, 21, 29, 37, 53, 69, 101, 85, 117, 181, 245, 309, 437, 565, 821, 693, 949, 1205, 1717, 1461, 1973, 2741, 3765, 2485, 3509, 5557, 7605, 9653, 13749, 17845, 26037, 21941, 30133, 38325, 54709, 46517, 62901, 87477, 120245, 79285, 112053, 144821

Offset: 1

Antti Karttunen, Nov 16 2012

The binary code is the same as used by function general-car-cdr of MIT/GNU Scheme: a zero bit represents a cdr operation (taking the right hand side branch in the binary tree), and a one bit represents a car (taking the left hand side branch in the binary tree). The bits are interpreted from LSB to MSB, and the most significant one bit, rather than being interpreted as an operation, signals the end of the binary code.

			As we must traverse to 4 in A218778-tree (see the example there) by first taking the left branch (car) from the root, resulting bit 1 as the least significant bit of the code, then by taking the right branch (cdr) from 3 to get to 4, resulting bit 0 as the second rightmost bit of the code, which when capped with an extra termination-one, results binary code 101, 5 in decimal, thus a(4)=5.

A. Karttunen, Table of n, a(n) for n = 1..1024
MIT/GNU Scheme 9.1. documentation, Function general-car-cdr

a(n) = A054429(A218615(n)). Superset of A218790. Used to construct A218778, A218779. Cf. also A218787, A218788

a(1)=1; for even n, a(n) = A004754(a(A011371(n))); for odd n, a(n) = A004755(a(A011371(n))).

A218776 A014486-codes for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the left hand side" construction.

Original entry on oeis.org

2, 12, 50, 204, 818, 3298, 13202, 52834, 211346, 845586, 3382418, 13531282, 54125714, 216503058, 866012306, 3464049426, 13856197778, 55424792722, 221699171474, 886796698770, 3547186799762, 14188747200658, 56754988803218, 227019955225746, 908079820907666

Offset: 1

Antti Karttunen, Nov 17 2012

nonn

The active middle region of the triangle (see attached "Wolframesque" illustration) corresponds to the area where the growing tip(s) of the beanstalk are located. Successively larger "turbulences" occurring in that area appear approximately at the row numbers given by A218548. The larger tendrils (the finite side-trees) are, the longer there is vacillation in the direction of the growing region, which lasts until the growing tip of the infinite stem (A179016) has passed the topmost tips of the tendril. See also A218612.

These are the mirror-images (in binary tree sense) of the terms in sequence A218778. For more compact versions, see A218780 & A218782.

			Illustration how the growing beanstalk-tree produces the first four terms of this sequence. In this variant, the lesser numbers come to the left hand side:
..........
...\1/.... Coded by A014486(A218777(1)) = A014486(1) = 2 (binary 10).
..........
..........
.\2/......
...\1/.... Coded by A014486(A218777(2)) = A014486(3) = 12 (bin. 1100).
..........
..........
.\2/ \3/..
...\1/.... Coded by A014486(A218777(3)) = A014486(6) = 50 (110010).
..........
..........
....\4/...
.\2/.\3/..
...\1/.... Coded by A014486(A218777(4)) = A014486(15) = 204 (11001100).
..........
Thus the first four terms of this sequence are 2, 12, 50 and 204.

Antti Karttunen, Table of n, a(n) for n = 1..256
Antti Karttunen, Terms a(1)-a(4096) drawn as binary strings, in Wolframesque fashion.

a(n) = A014486(A218777(n)). Cf. A014486, A218615, A218780, A218782, A218787, A218778.

A218779 A014486-indices for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the right hand side" construction.

Original entry on oeis.org

1, 2, 6, 16, 53, 156, 491, 1584, 5286, 17855, 61405, 214012, 754731, 2688307, 9659023, 34947356, 127260566, 465920651, 1714294995, 6335423798, 23506865260, 87534068725, 327024845325, 1225395355165, 4604242172697, 17343284212635, 65480496260109, 247755060623522

Offset: 1

Antti Karttunen, Nov 17 2012

nonn

See comments and examples at A218778.

Antti Karttunen, Table of n, a(n) for n = 1..128

Cf. A218778, A218781, A218783.

a(n) = A080300(A218778(n)).

a(n) = A057163(A218777(n)).

A218780 A014486-codes for the compact representation of Beanstalk-tree, growing by two natural numbers at time, starting from the tree of one internal node (1) and two leaves (2 and 3), with the lesser numbers coming to the left hand side.

Original entry on oeis.org

2, 10, 44, 180, 728, 2928, 11720, 46888, 187568, 750304, 3001232, 12004960, 48019856, 192079504, 768318048, 3073272224, 12293088960, 49172355968, 196689423936, 786757695872, 3147030783552, 12588123134528, 50352492538240, 201409970153216, 805639880612992

Offset: 1

Antti Karttunen, Nov 17 2012

nonn

The active middle region of the triangle (see the attached "Wolframesque" illustration) corresponds to the area where the growing tip(s) of the beanstalk are located. Successively larger "turbulences" occurring in that area appear approximately at the row numbers given by A218548 divided by two. The larger tendrils, (the finite side-trees) are, the longer there is vacillation in the direction of the growing region, which lasts until the growing tip of the infinite stem (A179016) has passed the topmost tips of the tendril. See also A218612.

These are the mirror-images (in binary tree sense) of the terms in sequence A218782. For less compact versions, see A218776 & A218778.

			Illustration how the growing beanstalk-tree produces the first four terms of this sequence. In this "compact" variant, each successive pair of numbers ((2,3), (4,5), (6,7), etc.) adds a new bud (\/) to the beanstalk, with the lesser numbers coming to the left hand side:
----------
..2...3...
...\./.... Coded by A014486(A218781(1)) = A014486(1) = 2 (binary 10).
....1.....
----------
....4...5.
.....\./..
..2...3...
...\./.... Coded by A014486(A218781(2)) = A014486(2) = 10 (bin. 1010).
....1.....
----------
..6...7...
...\./....
....4...5.
.....\./..
..2...3...
...\./.... Coded by A014486(A218781(3)) = A014486(5) = 44 (101100).
....1.....
----------
....8...9.
.....\./..
..6...7...
...\./....
....4...5.
.....\./..
..2...3...
...\./.... Coded by A014486(A218781(4)) = A014486(12) = 180 (10110100).
....1.....
----------
Thus the first four terms of this sequence are 2, 10, 44 and 180.

A. Karttunen, Table of n, a(n) for n = 1..256
A. Karttunen, Terms a(1)-a(4096) drawn as binary strings, in Wolframesque fashion.

a(n) = A014486(A218781(n)). Cf. A014486, A218791, A218776, A218778, A218782, A218787.

A218782 A014486-codes for the compact representation of Beanstalk-tree, growing by two natural numbers at time, starting from the tree of one internal node (1) and two leaves (3 and 2), with the larger numbers coming to the left hand side.

Original entry on oeis.org

2, 12, 52, 216, 872, 3496, 14024, 56200, 224904, 899720, 3599496, 14398600, 57599112, 230398088, 921606280, 3686471816, 14745933960, 58983782536, 235935438984, 943742064776, 3774970665096, 15099883493512, 60399541098632, 241598171519112, 966392760309896

Offset: 1

Antti Karttunen, Nov 17 2012

nonn

The active middle region of the triangle (see attached "Wolframesque" illustration) corresponds to the area where the growing tip(s) of the beanstalk are located. Successively larger "turbulences" occurring in that area appear approximately at the row numbers given by A218548 divided by two. The larger tendrils, (the finite side-trees) are, the longer there is vacillation in the direction of the growing region, which lasts until the growing tip of the infinite stem (A179016) has passed the topmost tips of the tendril. See also A218612.

These are the mirror-images (in binary tree sense) of the terms in sequence A218780. For less compact versions, see A218778 & A218776.

			Illustration how the growing beanstalk-tree produces the first four terms of this sequence. In this "compact" variant, each successive pair of numbers ((2,3), (4,5), (6,7), etc.) adds a new bud (\/) to the beanstalk, with the lesser numbers coming to the right hand side:
----------
..3...2...
...\./.... Coded by A014486(A218783(1)) = A014486(1) = 2 (binary 10).
....1.....
----------
5...4.....
.\./......
..3...2...
...\./.... Coded by A014486(A218783(2)) = A014486(3) = 12 (bin. 1100).
....1.....
----------
..7...6...
...\./....
5...4.....
.\./......
..3...2...
...\./.... Coded by A014486(A218783(3)) = A014486(7) = 52 (110100).
....1.....
----------
9...8.....
.\./......
..7...6...
...\./....
5...4.....
.\./......
..3...2...
...\./.... Coded by A014486(A218783(4)) = A014486(18) = 216 (11011000).
....1.....
----------
Thus the first four terms of this sequence are 2, 12, 52 and 216.

A. Karttunen, Table of n, a(n) for n = 1..256
A. Karttunen, Terms a(1)-a(4096) drawn as binary strings, in Wolframesque fashion.

a(n) = A014486(A218783(n)). Cf. A014486, A218790, A218778, A218776, A218780, A218788.

cp's OEIS Frontend

A218614 a(n) = binary code (shown here in decimal) of the position of natural number n in the beanstalk-tree A218778.

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A218776 A014486-codes for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the left hand side" construction.

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A218779 A014486-indices for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the right hand side" construction.

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A218780 A014486-codes for the compact representation of Beanstalk-tree, growing by two natural numbers at time, starting from the tree of one internal node (1) and two leaves (2 and 3), with the lesser numbers coming to the left hand side.

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A218782 A014486-codes for the compact representation of Beanstalk-tree, growing by two natural numbers at time, starting from the tree of one internal node (1) and two leaves (3 and 2), with the larger numbers coming to the left hand side.

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Comments

Examples

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