A255122 -id:A255122 - cp's OEIS frontend

A255056 Trunk of number-of-runs beanstalk: The unique infinite sequence such that a(n-1) = a(n) - number of runs in binary representation of a(n).

0, 2, 4, 6, 10, 12, 14, 18, 22, 26, 28, 30, 32, 36, 42, 46, 50, 54, 58, 60, 62, 64, 68, 74, 78, 84, 90, 94, 96, 100, 106, 110, 114, 118, 122, 124, 126, 128, 132, 138, 142, 148, 152, 156, 162, 168, 174, 180, 186, 190, 192, 196, 202, 206, 212, 218, 222, 224, 228, 234, 238, 242, 246, 250, 252, 254

Offset: 0

Antti Karttunen, Feb 14 2015

All numbers of the form (2^n)-2 are present, which guarantees the uniqueness and also provides a well-defined method to compute the sequence, for example, via a partially reversed version A255066.

The sequence was inspired by a similar "binary weight beanstalk", A179016, sharing some general properties with it (like its partly self-copying behavior, see A255071), but also differing in some aspects. For example, here the branching degree is not the constant 2, but can vary from 1 to 4. (Cf. A255058.)

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

First differences: A255336.
Terms halved: A255057.
Cf. A255053 & A255055 (the lower & upper bound for a(n)) and also A255123, A255124 (distances to those limits).
Cf. A255327, A255058 (branching degree for node n), A255330 (number of nodes in the finite subtrees branching from the node n), A255331, A255332
Subsequence: A000918 (except for -1).
Cf. A255061, A255062, A255071, A255072, A255066, A255122.
Cf. A254113, A254114.
Cf. A255063, A255064, A255125, A255126.
Similar "beanstalk's trunk" sequences using some other subtracting map than A236840: A179016, A219648, A219666.

(define (A255056 n) (A255066 (A255122 n)))

a(n) = A255066(A255122(n)).
Other identities and observations. For all n >= 0:
a(n) = 2*A255057(n).
A255072(a(n)) = n.
A255053(n) <= a(n) <= A255055(n).

A255057 The trunk of number-of-runs beanstalk, halved: a(n) = A255056(n)/2.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 16, 18, 21, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 42, 45, 47, 48, 50, 53, 55, 57, 59, 61, 62, 63, 64, 66, 69, 71, 74, 76, 78, 81, 84, 87, 90, 93, 95, 96, 98, 101, 103, 106, 109, 111, 112, 114, 117, 119, 121, 123, 125, 126, 127

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

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

First differences: A255337.
Characteristic function: A255339.
Cf. A255056, A255058, A255067, A255122, A255125, A255126, A255328, A255329, A255330, A255338.

a(n) = A255056(n)/2.

a(n) = A255067(A255122(n)).

A255066 The trunk of number-of-runs beanstalk (A255056) with reversed subsections.

Original entry on oeis.org

0, 2, 6, 4, 14, 12, 10, 30, 28, 26, 22, 18, 62, 60, 58, 54, 50, 46, 42, 36, 32, 126, 124, 122, 118, 114, 110, 106, 100, 96, 94, 90, 84, 78, 74, 68, 64, 254, 252, 250, 246, 242, 238, 234, 228, 224, 222, 218, 212, 206, 202, 196, 192, 190, 186, 180, 174, 168, 162, 156, 152, 148, 142, 138, 132, 128, 510

Offset: 0

Antti Karttunen, Feb 14 2015

This can be viewed as an irregular table: after the initial zero on row 0, start each row n with term x = (2^(n+1))-2 and subtract repeatedly the number of runs in binary representation of x to get successive x's, until the number that has already been listed (which is always (2^n)-2) is encountered, which is not listed second time, but instead, the current row is finished [and thus containing only terms of equal binary length, A000523(n) on row n]. The next row then starts with (2^(n+2))-2, with the same process repeated.

			Rows 0 - 5 of the array:
0;
2;
6, 4;
14, 12, 10;
30, 28, 26, 22, 18;
62, 60, 58, 54, 50, 46, 42, 36, 32;
After row 0, the length of row n is given by A255071(n).

Antti Karttunen, Table of n, a(n) for n = 0..16142; Rows 0..16, flattened

Cf. A255067 (same seq, terms divided by 2).
Cf. A255071 (gives row lengths).
Cf. A000523, A004755, A005811, A236840, A255056, A255122.
Analogous sequences: A218616, A230416.

a(0) = 0, a(1) = 2, a(2) = 6; and for n > 2, a(n) = A004755(A004755(A236840(a(n-1)))) if A236840(a(n-1))+2 is power of 2, otherwise just A236840(a(n-1)) [where A004755(x) adds one 1-bit to the left of the most significant bit of x].
In other words, for n > 2, let k = A236840(a(n-1)). Then, if k+2 is not a power of 2, a(n) = k, otherwise a(n) = k + (6 * (2^A000523(k))).
Other identities. For all n >= 0:
a(n) = A255056(A255122(n)).

A255067 Terms of A255066 halved.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 5, 15, 14, 13, 11, 9, 31, 30, 29, 27, 25, 23, 21, 18, 16, 63, 62, 61, 59, 57, 55, 53, 50, 48, 47, 45, 42, 39, 37, 34, 32, 127, 126, 125, 123, 121, 119, 117, 114, 112, 111, 109, 106, 103, 101, 98, 96, 95, 93, 90, 87, 84, 81, 78, 76, 74, 71, 69, 66, 64, 255

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

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

Cf. A255066, A255057, A255122, A255125, A255126.

Scheme
```
(define (A255067 n) (/ (A255066 n) 2))
```

a(n) = A255066(n)/2.
Other identities. For all n >= 0:
a(n) = A255057(A255122(n)).

A255121 After zero, each n occurs A255071(n) times.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

An auxiliary sequence for computing A255122 and A255056.

Antti Karttunen, Table of n, a(n) for n = 0..721

Cf. A255056, A255061, A255071, A255122.

Similar sequence: A213711.

A255120 After the first zero, numbers from 0 to A255071(n)-1 followed by numbers from 0 to A255071(n+1)-1, etc.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

An auxiliary sequence for computing A255122 and A255056.

Antti Karttunen, Table of n, a(n) for n = 0..721

Cf. A255056, A255062, A255121, A255122.

Similar sequence: A218601.

(define (A255120 n) (if (zero? n) n (- n (A255062 (A255121 n)))))

a(0) = 0; and for n >= 1, a(n) = n - A255062(A255121(n)).

cp's OEIS Frontend

A255056 Trunk of number-of-runs beanstalk: The unique infinite sequence such that a(n-1) = a(n) - number of runs in binary representation of a(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A255057 The trunk of number-of-runs beanstalk, halved: a(n) = A255056(n)/2.

Views

Author

Keywords

Links

Crossrefs

Formula

A255066 The trunk of number-of-runs beanstalk (A255056) with reversed subsections.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A255067 Terms of A255066 halved.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A255121 After zero, each n occurs A255071(n) times.

Views

Author

Keywords

Comments

Links

Crossrefs

A255120 After the first zero, numbers from 0 to A255071(n)-1 followed by numbers from 0 to A255071(n+1)-1, etc.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula