A218604 -id:A218604 - cp's OEIS frontend

A218608 The positions of zeros in A218604, i.e., those integers i for which A179016(i) = A173601(i).

0, 1, 2, 4, 6, 7, 9, 10, 12, 14, 15, 21, 23, 24, 38, 40, 41, 61, 62, 63, 64, 68, 70, 71, 91, 92, 93, 94, 104, 105, 106, 107, 113, 122, 124, 125, 145, 146, 147, 148, 158, 159, 160, 161, 167, 182, 183, 184, 185, 191, 202, 220, 222, 223, 243, 244, 245, 246, 256

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 greatest possible position of the allotted "window" which it at that point must pass through, i.e. there are no leaves or side-trees at its right (greater) side at these points. (See comments at A218604.)

Antti Karttunen, Table of n, a(n) for n = 0..7730

Apart from zero, a subset of A213733. Cf. A218607, A218606.

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

A173601 Greatest inverse of A071542, i.e., a(n) = maximal i such that A071542(i) = n.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 15, 17, 19, 23, 27, 31, 33, 35, 39, 43, 47, 51, 55, 59, 63, 65, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 125, 127, 129, 131, 135, 139, 143, 147, 151, 155, 159, 163, 167, 171, 175, 179, 183, 189, 191, 195, 199, 203, 207

Offset: 0

Charles R Greathouse IV, Oct 30 2012

nonn

What is s = lim sup a(n)/(n log_2(n))? A counting argument suggests s >= 1/2, and in any case s <= 1.

Essentially also the partial sums of A086876. - Antti Karttunen, Nov 10 2012 (per personal mail from Carl R. White, Nov 02 2012)

Charles R Greathouse IV and Antti Karttunen, Table of n, a(n) for n = 0..10000 (terms 1-10000 from Greathouse, Karttunen recomputed with prepended a(0)=0)

See A213708 for the least inverse. A086876 gives the first differences. Also, a(n)=A213708(n)+A086876(n)-1. Cf. A071542, A179016, A218604, A218608.

v=vectorsmall(10^3);v[1]=1;for(n=2,#v,v[n]=v[n-hammingweight(n)]+1); u=vector(solve(x=1,#v,x*log(x)/log(2)-#v)\1);for(i=1,#v,if(v[i]<=#u,u[v[i]]=i)); u

;; With Antti Karttunen's intseq-library:
(define A173601 (PARTIALSUMS 1 0 (compose-funs A086876 1+)))

a(n)/log_2(a(n)) < n < a(n) for n > 1.

Changed the starting offset by prepending a(0)=0 (with the indexing of the rest of terms thus not changed) - Antti Karttunen, Nov 10 2012

A218603 a(n) = A179016(n) - A213708(n).

Original entry on oeis.org

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

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 lesser 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: A218607, A218605.

Cf. A086876, A179016, A213708, A218604.

(define (A218603 n) (- (A179016 n) (A213708 n)))

a(n) = A086876(n) - A218604(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.

A255124 a(n) = A255055(n) - A255056(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

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

Cf. A255055, A255056, A255123.

Analogous sequence: A218604.

(define (A255124 n) (- (A255055 n) (A255056 n)))

a(n) = A255055(n) - A255056(n).

A218606 a(n) = A218608(n-1) + 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 22, 24, 25, 39, 41, 42, 62, 63, 64, 65, 69, 71, 72, 92, 93, 94, 95, 105, 106, 107, 108, 114, 123, 125, 126, 146, 147, 148, 149, 159, 160, 161, 162, 168, 183, 184, 185, 186, 192, 203, 221, 223, 224, 244, 245, 246, 247, 257

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 greatest possible position of the allotted "window" which it at that point must pass through. (See comments at A218604.)

Antti Karttunen, Table of n, a(n) for n = 1..17447

Cf. A218604, A218605, A218608.

(define (A218606 n) (1+ (A218608 (-1+ n))))

a(n) = A218608(n-1) + 1.

Definition changed because of the changed offset of A218604. - Antti Karttunen, Nov 10 2012

A231724 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the greatest integer (A219655(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219655(n) - A219666(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 3, 2, 0, 0, 1, 3, 2, 1, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 6, 0, 0, 1, 3, 2, 1, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 7, 4, 4, 5, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 7, 5, 7, 7, 5, 6, 6, 1, 3, 4, 4, 3, 4, 5, 7, 5, 7, 7, 5, 6, 6, 2, 2, 3, 4, 5

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 right (greater) 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 A218604 relates to the binary system and similar remarks apply here.

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

Cf. A231723, A230409, A219662 & A219663.

(define (A231724 n) (- (A219655 n) (A219666 n)))

a(n) = A219655(n) - A219666(n).

A219654(n) = a(n) + A231723(n) + 1.

cp's OEIS Frontend

A218608 The positions of zeros in A218604, i.e., those integers i for which A179016(i) = A173601(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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A173601 Greatest inverse of A071542, i.e., a(n) = maximal i such that A071542(i) = n.

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A218603 a(n) = A179016(n) - A213708(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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A255124 a(n) = A255055(n) - A255056(n).

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A218606 a(n) = A218608(n-1) + 1.

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A231724 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the greatest integer (A219655(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219655(n) - A219666(n).