A255071 -id:A255071 - cp's OEIS frontend

A255122 Simple self-inverse permutation of natural numbers: after zero, list each block of A255071(n) numbers in reverse order, from A255061(n+1) to A255062(n).

0, 1, 3, 2, 6, 5, 4, 11, 10, 9, 8, 7, 20, 19, 18, 17, 16, 15, 14, 13, 12, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 118

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Maps between A255056 and A255066. (Equally, between A255057 and A255067.)

Antti Karttunen, Table of n, a(n) for n = 0..721
Index entries for sequences that are permutations of the natural numbers

Cf. A255061, A255062, A255071, A255120, A255121.
Cf. A255056, A255057, A255066, A255067.
Similar permutations: A218602, A230432.

(define (A255122 n) (- (A255061 (+ 1 (A255121 n))) (A255120 n)))

a(n) = A255061(1+A255121(n)) - A255120(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)).

A254119 a(n) = A213709(n) - A255071(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 126, 204, 303, 414, 526, 652, 907, 1705, 4213, 11329, 29670, 73408, 171345, 379494, 802875, 1632745, 3210086, 6134908, 11457641, 21020580, 38076142

Offset: 1

Antti Karttunen, Feb 15 2015

nonn

Cf. A213709, A255071.

(define (A254119 n) (- (A213709 n) (A255071 n)))

a(n) = A213709(n) - A255071(n).

A255069 First differences of A255071.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 44, 81, 150, 280, 526, 992, 1875, 3551, 6740, 12823, 24450, 46709, 89383, 171325, 328962, 632849, 1219909, 2356217, 4559224, 8835610, 17144046, 33295497, 64705083, 125802338, 244673791, 476011284, 926373373, 1803512210, 3512774806

Offset: 1

Antti Karttunen, Feb 21 2015

nonn

Also, a(n) = the number of times a number whose binary expansion begins with 10... (cf. A004754) is encountered when iterating from 2^(n+2)-2 to (2^(n+1))-2 with the map x -> x - (number of runs in binary representation of x), i.e., with m(n) = A236840(n). For example, when starting from the initial value (2^(4+2))-2 = 62, we get m(62) = 60, m(60) = 58, m(58) = 54, m(54) = 50, m(50) = 46, m(46) = 42, m(42) = 36 and finally m(36) = 32, which is (2^(4+1)). Of the nine numbers encountered, only 46, 42, 36 and 32 (in binary: 101110, 101010, 100100 and 100000) are in A004754, thus a(4) = 5.

First differences of A255071.
Cf. A004754, A005811, A236840, A255061, A255062, A255056, A255066.
Cf. also A255063, A255064, A255125, A255126.
Analogous sequence: A226060.

(define (A255069 n) (- (A255071 (+ n 1)) (A255071 n)))
(define (A255069shifted n) (add (lambda (n) (- 1 (A079944off2 (A255066 n)))) (A255062 n) (A255061 (+ 1 n)))) ;; Cf. A079444.

a(n) = A255071(n+1) - A255071(n).
For n > 1, a(n-1) = Sum_{k = A255062(n) .. A255061(n+1)}(1-secondmsb(A255056(k))).
Here secondmsb is implemented by the starting offset 2 version of A079944, and effectively gives the second most significant bit in the binary expansion of n. The formula follows from the semi-regular nature of number-of-runs beanstalk, see comments above and at A255071.

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).

Original entry on oeis.org

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).

A236840 n minus number of runs in the binary expansion of n: a(n) = n - A005811(n).

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 4, 6, 6, 6, 6, 8, 10, 10, 12, 14, 14, 14, 14, 16, 16, 16, 18, 20, 22, 22, 22, 24, 26, 26, 28, 30, 30, 30, 30, 32, 32, 32, 34, 36, 36, 36, 36, 38, 40, 40, 42, 44, 46, 46, 46, 48, 48, 48, 50, 52, 54, 54, 54, 56, 58, 58, 60, 62, 62, 62, 62, 64, 64, 64

Offset: 0

Antti Karttunen, Apr 18 2014

All terms are even. Used by the "number-of-runs beanstalk" sequence A255056 and many of its associated sequences.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A091067 (the positions of records), A106836 (run lengths).
Cf. A255070 (terms divided by 2).
Cf. A005811, A000120, A003188.
Cf. A255056, A255066, A255061, A255062, A255071, A255072, A255058, A255327.
Other subtracting maps: A011371, A178503, A219641, A219651, A227190, A227191, A237449, A244320, A244234, A255131.

A236840 := proc(n) local i, b; if n=0 then 0 else b := convert(n, base, 2); select(i -> (b[i-1]<>b[i]), [$2..nops(b)]); n-1-nops(%) fi end: seq(A236840(i), i=0..69); # Peter Luschny, Apr 19 2014

a[n_] := n - Length@ Split[IntegerDigits[n, 2]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 16 2023 *)

Scheme
```
(define (A236840 n)  (- n (A005811 n)))
```

a(n) = n - A005811(n) = n - A000120(A003188(n)).

a(n) = 2*A255070(n).

A255061 Number of steps to reach 0 when starting from (2^n)-2 and iterating the map x -> x - (number of runs in binary representation of x): a(n) = A255072(A000918(n)).

Original entry on oeis.org

0, 1, 3, 6, 11, 20, 36, 65, 118, 215, 393, 721, 1329, 2463, 4589, 8590, 16142, 30434, 57549, 109114, 207388, 395045, 754027, 1441971, 2762764, 5303466, 10200385, 19656528, 37948281, 73384080, 142115376, 275551755, 534790472, 1038702980, 2018626772, 3924923937, 7634733312

Offset: 1

Antti Karttunen, Feb 14 2015

nonn

Apart from a(1)=1, also gives the positions of ones in A255054.

One less than A255062.
First differences: A255071.
Apart from a(1)=1, a subsequence of A255059.
Cf. A000918, A255072, A255054.
Analogous sequences: A218600, A226061.

(define (A255061 n) (A255072 (A000918 n)))
(define (A255061 n) (if (= 1 n) 0 (+ (A255061 (- n 1)) (A255071 (- n 1))))) ;; Assuming that A255071 has been already computed, with e.g. the PARI-program given in that entry.

a(n) = A255072(A000918(n)).
a(1) = 0; for n > 1, a(n) = a(n-1) + A255071(n-1).
Other identities. For all n >= 1:
a(n) = A255062(n) - 1.

A255062 Number of steps to reach 0 when starting from (2^n)-1 and iterating the map x -> x - (number of runs in binary representation of x): a(n) = A255072(A000225(n)).

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 21, 37, 66, 119, 216, 394, 722, 1330, 2464, 4590, 8591, 16143, 30435, 57550, 109115, 207389, 395046, 754028, 1441972, 2762765, 5303467, 10200386, 19656529, 37948282, 73384081, 142115377, 275551756, 534790473, 1038702981, 2018626773, 3924923938, 7634733313

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Also, for n >= 1, the number of steps to reach 0 when starting from 2^n and iterating the map x -> x minus A005811(x), the number of runs in binary representation of x.

One more than A255061.
First differences: A255071 (after the zero term).
Cf. A000079, A000225, A005811, A255071, A255072.
Analogous sequences: A213710 (A218600), A219665.

(define (A255062 n) (A255072 (A000225 n)))
(define (A255062 n) (if (<= n 1) n (+ (A255062 (- n 1)) (A255071 (- n 1))))) ;; Assuming that A255071 has been already computed, with e.g. the PARI-program given in that entry.

a(n) = A255072(A000225(n)).
a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) + A255071(n-1).
Other identities. For all n >= 1:
a(n) = A255072(A000079(n)). [See the Comments section.]
a(n) = 1 + A255061(n).

A255072 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (number of runs in binary representation of x).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A255053 (least inverse), A255055 (greatest inverse), A255054 (run lengths).
Cf. A005811, A236840, A255071, A255056.
Cf. A255061 & A255062 (values at points (2^n)-2 and (2^n)-1).
Analogous sequences: A071542, A219642, A219652

a(0) = 0; for n >= 1, a(n) = 1 + a(A236840(n)) = 1 + a(n - A005811(n)).
Other identities. For all n >= 0:
a(A255053(n)) = a(A255055(n)) = n.
a(A255056(n)) = n. [This sequence works also as an inverse function for number-of-runs beanstalk A255056.]

cp's OEIS Frontend

A255122 Simple self-inverse permutation of natural numbers: after zero, list each block of A255071(n) numbers in reverse order, from A255061(n+1) to A255062(n).

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

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

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A254119 a(n) = A213709(n) - A255071(n).

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A255069 First differences of A255071.

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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).

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A236840 n minus number of runs in the binary expansion of n: a(n) = n - A005811(n).

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Mathematica

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A255061 Number of steps to reach 0 when starting from (2^n)-2 and iterating the map x -> x - (number of runs in binary representation of x): a(n) = A255072(A000918(n)).

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A255062 Number of steps to reach 0 when starting from (2^n)-1 and iterating the map x -> x - (number of runs in binary representation of x): a(n) = A255072(A000225(n)).

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