A218603 -id:A218603 - cp's OEIS frontend

A218607 The positions of zeros in A218603, i.e., those integers i for which A179016(i) = A213708(i).

0, 1, 3, 5, 8, 13, 22, 33, 35, 36, 37, 39, 50, 52, 53, 54, 55, 58, 69, 80, 82, 83, 84, 85, 88, 101, 118, 123, 134, 136, 137, 138, 139, 142, 155, 172, 179, 196, 207, 212, 221, 232, 234, 235, 236, 237, 240, 253, 270, 277, 294, 305, 310, 321, 338, 349, 354, 369, 374, 383, 400

Offset: 0

Antti Karttunen, Nov 10 2012

nonn

These are the points i for which the corresponding node in the infinite trunk of beanstalk (A179016(i)) is at the least possible position of its allotted "window" which it at that point must pass through, i.e., there are no leaves or side-trees at its left (lesser) side at these points. (See comments at A218603.)

Antti Karttunen, Table of n, a(n) for n = 0..7948

Apart from first two terms, a subset of A213732. Cf. A218608, A218605.

A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 11, 15, 16, 19, 23, 26, 31, 32, 35, 39, 42, 46, 49, 53, 57, 63, 64, 67, 71, 74, 78, 81, 85, 89, 94, 97, 101, 104, 109, 112, 116, 120, 127, 128, 131, 135, 138, 142, 145, 149, 153, 158, 161, 165, 168, 173, 176, 180, 184, 190, 193, 197, 200, 205, 209

Offset: 0

Carl R. White, Jun 24 2010

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "binary beanstalk" from its root (zero). The name "beanstalk" is due to Antti Karttunen.

There are many finite sequences such as 0,1,2; 0,1,3,4,7,9; etc. obeying the same condition (see A218254) and as the length increases, so (necessarily) does the similarity to this infinite sequence.

Alois P. Heinz and Antti Karttunen, Table of n, a(n) for n = 0..16405 (first 1000 terms from Alois P. Heinz)
Paul Tek, Illustration of the first terms

Cf. A000120, A010062, A011371, A213710, A213711, A213717, A213730, A213731, A218600, A218616, A218789, A233271, A218602, A054429. First differences: A213712, complement: A213713.
A subsequence of A005187, i.e., a(n) = A005187(A213715(n)). For all n,
A071542(a(n)) = n, and furthermore A213708(n) <= a(n) <= A173601(n). (Cf. A218603, A218604).
Rows of A218254, when reversed, converge towards this sequence.
Cf. A276623, A219648, A219666, A255056, A276573, A276583, A276613 for analogous constructions, and also A259934.

TakeWhile[Reverse@ NestWhileList[# - DigitCount[#, 2, 1] &, 10^3, # > 0 &], # <= 209 &] (* Michael De Vlieger, Sep 12 2016 *)

a(0)=0, a(1)=1, and for n > 1, if n = A218600(A213711(n)) then a(n) = (2^A213711(n)) - 1, and in other cases, a(n) = a(n+1) - A213712(n+1). (This formula is based on Carl White's observation that this iterated/converging path must pass through each (2^n)-1. However, it would be very interesting to know whether the sequence admits more traditional recurrence(s), referring to previous, not to further terms in the sequence in their definition!) - Antti Karttunen, Oct 26 2012
a(n) = A218616(A218602(n)). - Antti Karttunen, Mar 04 2013
a(n) = A054429(A233271(A218602(n))). - Antti Karttunen, Dec 12 2013

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

A218604 a(n) = A173601(n) - A179016(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 2, 2, 0, 1, 0, 0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 5, 0, 1, 0, 0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 5, 1, 2, 2, 3, 2, 2, 0, 0, 0, 0, 1, 3, 2, 0, 1, 0, 0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 5, 1, 2

Offset: 0

Antti Karttunen, Nov 03 2012

nonn

For all n, the following holds: A213708(n) <= A179016(n) <= A173601(n). This sequence gives the distance of the node n in the infinite trunk of beanstalk (A179016(n)) from the greater edge of the A086876(n) wide window which it at that point must pass through.

The increasing steps seem to be quite constrained in their magnitude, compared to the decreasing steps. (This depends on how the "tendrils",i.e. the finite side-trees on the other side of the infinite trunk grow and reach their tops).

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

Positions of zeros: A218608, A218606.

Cf. A086876, A173601, A179016, A218603.

(define (A218604 n) (- (A173601 n) (A179016 n)))

a(n) = A086876(n) - A218603(n) - 1.

Offset changed because of the changed offset of A179016 - Antti Karttunen, Nov 10 2012

A218789 Partial sums of A218618.

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 4, 3, 6, 11, 10, 11, 10, 13, 18, 17, 10, 11, 12, 15, 16, 15, 18, 23, 22, 15, 16, 17, 20, 13, 14, 15, 10, 11, 6, 3, 2, 3, 2, 5, 10, 9, 2, 3, 4, 7, 0, 1, 2, -3, -2, -7, -10, -11, -18, -17, -16, -21, -20, -9, -8, -7, 6, 7, 2, 1, 4, 5, 4, 7, 12

Offset: 0

Antti Karttunen, Dec 03 2012

The term a(n) indicates approximately the "balance" of the binary beanstalk (cf. A179016) at n steps up from the root, which in turn has repercussions for the sequences such as A218542 and A218543.

Antti Karttunen, Table of n, a(n) for n = 0..30901

Cf. A179016, A218618, A218603, A218604, A218543.

A218786 The sizes of the "tendrils" (finite side-trees sprouting at A213730, A218787) of infinite beanstalk (A179016).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 3, 0, 0, 1, 0, 0, 1, 2, 0, 3, 0, 0, 1, 3, 0, 0, 2, 0, 2, 1, 0, 0, 0, 1, 2, 0, 3, 0, 0, 1, 3, 0, 0, 2, 0, 2, 1, 0, 3, 0, 0, 2, 0, 5, 0, 0, 6, 0, 2, 0, 1, 0, 0, 1, 2, 0, 3, 0, 0, 1, 3, 0, 0, 2, 0, 2, 1, 0, 3, 0, 0, 2

Offset: 1

Antti Karttunen, Nov 11 2012

			The first four tendrils of the beanstalk sprout at 2, 5, 6 and 9, (the first four nonzero terms of A213730) which are all leaves (i.e., in A055938), thus the first four terms of this sequence are all 0's. The next term A213730(5)=10, which is not leaf, but branches to two leaf-branches (12 and 13, as with both we have: 12-A000120(12)=10 and 13-A000120(13)=10, and both 12 and 13 are found from A055938, so the tendril at 10 is a binary tree of one internal vertex (and two leaves), i.e., \/, thus a(5)=1.

Antti Karttunen, Table of n, a(n) for n = 1..8727

Equally, a(n) = A072643(A218787(n)) = A072643(A218788(n)). Cf. A218613, A218603, A218604.

(define (A218786 n) (-1+ (A213726 (A213730 n))))

a(n) = A213726(A213730(n))-1.

A255123 a(n) = A255056(n) - A255053(n).

Original entry on oeis.org

0, 1, 1, 0, 3, 1, 0, 3, 3, 3, 1, 0, 1, 1, 3, 2, 3, 3, 3, 1, 0, 1, 1, 3, 2, 5, 4, 3, 1, 1, 3, 2, 3, 3, 3, 1, 0, 1, 1, 3, 2, 5, 2, 1, 3, 2, 2, 4, 4, 3, 1, 1, 3, 2, 5, 4, 3, 1, 1, 3, 2, 3, 3, 3, 1, 0, 1, 1, 3, 2, 5, 2, 1, 3, 2, 2, 4, 2, 1, 3, 2, 2, 3, 1, 2, 2, 2, 4, 4, 3, 1, 1, 3, 2, 5, 2, 1, 3, 2, 2, 4, 4, 3, 1, 1, 3, 2, 5, 4, 3, 1, 1, 3, 2, 3, 3, 3, 1, 0

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16142

Cf. A255053, A255056, A255061, A255124.

Analogous sequence: A218603.

(define (A255123 n) (- (A255056 n) (A255053 n)))

a(n) = A255056(n) - A255053(n).

A218605 a(n) = A218607(n-1) + 1.

Original entry on oeis.org

1, 2, 4, 6, 9, 14, 23, 34, 36, 37, 38, 40, 51, 53, 54, 55, 56, 59, 70, 81, 83, 84, 85, 86, 89, 102, 119, 124, 135, 137, 138, 139, 140, 143, 156, 173, 180, 197, 208, 213, 222, 233, 235, 236, 237, 238, 241, 254, 271, 278, 295, 306, 311, 322, 339, 350, 355, 370, 375

Offset: 1

Antti Karttunen, Nov 03 2012

nonn

These are the points i for which the predecessor node (i.e. the vertex that is one step towards the root) in the infinite trunk of beanstalk (A179016(i)) is at the least possible position of its allotted "window" which it at that point must pass through. (See comments at A218603 and A218607.)

Antti Karttunen, Table of n, a(n) for n = 1..7949

Cf. A218603, A218606, A218607.

(define (A218605 n) (1+ (A218607 (-1+ n))))

a(n) = A218607(n-1) + 1.

Definition changed because of the changed offset of A218603. - Antti Karttunen, Nov 10 2012

A231723 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the smallest integer (A219653(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219666(n) - A219653(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 1, 3, 1, 2, 0, 1, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 1, 3, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 5, 7, 8, 1, 0

Offset: 0

Antti Karttunen, Nov 13 2013

nonn

For all n, the following holds: A219653(n) <= A219666(n) <= A219655(n). This sequence gives the distance of the node n in the infinite trunk of factorial beanstalk (A219666(n)) from the left (lesser) edge of the A219654(n) wide window which it at that point must pass through.

This sequence relates to the factorial base representation (A007623) in the same way as A218603 relates to the binary system and similar remarks apply here.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. A231724, A230409, A219662 & A219663.

(define (A231723 n) (- (A219666 n) (A219653 n)))

a(n) = A219666(n) - A219653(n).

A219654(n) = a(n) + A231724(n) + 1.

cp's OEIS Frontend

A218607 The positions of zeros in A218603, i.e., those integers i for which A179016(i) = A213708(i).

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A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

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A218604 a(n) = A173601(n) - A179016(n).

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A218789 Partial sums of A218618.

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A218786 The sizes of the "tendrils" (finite side-trees sprouting at A213730, A218787) of infinite beanstalk (A179016).

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A255123 a(n) = A255056(n) - A255053(n).

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A218605 a(n) = A218607(n-1) + 1.

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A231723 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the smallest integer (A219653(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219666(n) - A219653(n).

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