A218548 -id:A218548 - cp's OEIS frontend

A218550 a(n) = A213725(A218548(n)).

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 19, 15, 16, 27, 39, 51, 67

Offset: 1

Antti Karttunen, Nov 04 2012

a(n) tells the maximum depth + 1 of the n-th record size finite side-tree in the infinite beanstalk.

			a(7)=6 as the finite tree of 7 leaves (233, 234, 235, 238, 245, 250 and 251) and 6 internal vertices (244, 239, 232, 229, 228 and 224) rooted at 224 has the maximum depth of 5 (the path 224 -> 228 -> 232 -> 239 -> 244 -> 250/251).

Cf. A218549.

(define (A218550 n) (A213725 (A218548 n)))

A213730 After zero, gives the numbers where finite "side-trees" (or "tendrils") of beanstalk branch off from its infinite trunk (A179016).

Original entry on oeis.org

0, 2, 5, 6, 9, 10, 14, 17, 18, 22, 27, 30, 33, 34, 38, 43, 47, 48, 52, 56, 62, 65, 66, 70, 75, 79, 80, 84, 88, 95, 96, 100, 105, 108, 113, 117, 121, 126, 129, 130, 134, 139, 143, 144, 148, 152, 159, 160, 164, 169, 172, 177, 181, 185, 191, 192, 196, 201, 204, 208, 214, 220, 224, 230, 237, 241, 246, 254, 257, 258, 262, 267, 271, 272, 276, 280, 287, 288, 292, 297, 300, 305, 309, 313, 319, 320, 324

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

These are the other branches branching off from the infinite trunk, than the ones that stay in that infinite trunk. These other branches are always finite, except for the zero in which case the other branch leads to 1, and the other branch (listed here) leads back to zero itself (as 0-A000120(0) = 0).

The terms include also those leaves (i.e. those terms of A055938) that are right next to the infinite trunk.

Antti Karttunen, Table of n, a(n) for n = 0..8727

Cf. A218548.

: (define (A213730 n) (+ (A213723 (A179016 n)) (A213728 (1+ n))))
;; Alternative definition:
(define (A213730v2 n) (+ (A179016 (1+ n)) (expt -1 (A213729 (1+ n)))))

a(n) = A213723(A179016(n)) + A213728(n+1).

a(n) = A179016(n+1) + (-1)^A213729(n+1).

Offset changed from 1 to 0 by Antti Karttunen, Nov 05 2012

A213726 a(n)=0 if n is in the infinite trunk of the "beanstalk" (i.e., in A179016), otherwise number of terminal nodes (leaves) in that finite branch of the beanstalk.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 0, 3, 1, 1, 1, 0, 2, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 3, 0, 1, 2, 0, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 0, 3, 1, 1, 1, 0, 2, 1, 0, 3, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 0, 4, 1, 0, 3, 1, 1, 0, 2, 1, 0, 3, 1, 1, 1, 0, 2, 1, 6

Offset: 0

Antti Karttunen, Nov 01 2012

a(n) tells for each natural number n, whether it belongs to the infinite trunk of beanstalk (when a(n)=0), or if it is one of the terminal nodes (i.e., leaves, A055938) (when a(n)=1), or otherwise, when a(n)>1, tells from how many terminal nodes one can end to this n, by repeatedly subtracting their bit count (A000120) from them (as explained in A071542).

			a(10)=2 because the only numbers in A055938 from which one can end to 10 by the process described in A071542/A179016 are 12 and 13 (see comment at A213717). Similarly, a(22)=3 as there are following three cases: 24 as 24-A000120(24) = 24-2 = 22, and also 28 & 29 as 28-A000120(28) = 28-3 = 25, and 29-A000120(29) = 29-4 = 25, and then 25-A000120(25) = 25-3 = 22.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Differs from A213725 for the first time at n=208, where a(n)=6, while A213725(208)=5.

For all n, a(A179016(n)) = 0, a(A055938(n)) = 1, and a(A213717(n)) >= 2. For all n, A213727(A213717(n)) = (2*a(A213717(n)))-1. Cf. A213725-A213731. Records: A218548, A218549.

If A079559(n)=0, a(n)=1; otherwise, if A213719(n)=1, a(n)=0; otherwise a(n) = a(A213723(n))+a(A213724(n)).

A218612 Numbers which are the roots of distinct not-previously-encountered side-trees ("tendrils") sprouting from the side of the infinite beanstalk (see A213730).

Original entry on oeis.org

2, 10, 22, 47, 105, 208, 224, 471, 486, 943, 966, 974, 1934, 1972, 3509, 3546, 3765, 3893, 3930, 3995, 4027, 4049, 7912, 8041, 8058, 8146, 14291, 15315, 15738, 15827, 15995, 16040, 16122, 16211, 16312, 16334, 31694, 32207, 32440, 32462, 32568, 57145, 57208

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 1..72

A superset of A218548. Cf. A218611, A218613.

(define (A218612 n) (A213730 (A218611 n)))

a(n) = A213730(A218611(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.

A218778 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 right side" construction.

Original entry on oeis.org

2, 10, 50, 210, 914, 3666, 14738, 59026, 236690, 946834, 3787922, 15151762, 60607634, 242437266, 969821330, 3879357586, 15518026898, 62072179858, 248289315986, 993157336210, 3972629941394, 15890526653586, 63562180611218, 254248729332882, 1016994991328402

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 A218776. 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 right hand side:
..........
...\1/.... Coded by A014486(A218779(1)) = A014486(1) = 2 (binary 10).
..........
..........
.....\2/..
...\1/.... Coded by A014486(A218779(2)) = A014486(2) = 10 (bin. 1010).
..........
..........
.\3/ \2/..
...\1/.... Coded by A014486(A218779(3)) = A014486(6) = 50 (110010).
..........
..........
..\4/.....
.\3/.\2/..
...\1/.... Coded by A014486(A218779(4)) = A014486(16) = 210 (11010010).
..........
Thus the first four terms of this sequence are 2, 10, 50 and 210.

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(A218779(n)). Cf. A014486, A218614, A218776, A218782, A218780. A218788.

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.

A218549 Record values in A213726.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 10, 11, 18, 20, 33, 34, 35, 63, 144, 170, 241

Offset: 1

Antti Karttunen, Nov 04 2012

nonn

			A218548(7)=224, A213726(224)=7 thus a(7)=7 as there are the following seven leaves of beanstalk (i.e. terms of A055938): 233, 234, 235, 238, 245, 250 and 251 that will pass through 224, when an iterative process of x-A000120(x) (see A071542) is applied to them several times.

Cf. A218550.

(define (A218549 n) (A213726 (A218548 n)))

a(n) = A213726(A218548(n)).

cp's OEIS Frontend

A218550 a(n) = A213725(A218548(n)).

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A213730 After zero, gives the numbers where finite "side-trees" (or "tendrils") of beanstalk branch off from its infinite trunk (A179016).

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A213726 a(n)=0 if n is in the infinite trunk of the "beanstalk" (i.e., in A179016), otherwise number of terminal nodes (leaves) in that finite branch of the beanstalk.

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A218612 Numbers which are the roots of distinct not-previously-encountered side-trees ("tendrils") sprouting from the side of the infinite beanstalk (see A213730).

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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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A218778 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 right 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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A218549 Record values in A213726.

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