A255068 -id:A255068 - cp's OEIS frontend

A106836 First differences of A060833 and (from a(2) onward) also of A091067 and A255068.

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

Offset: 1

Ralf Stephan, May 03 2005

nonn

From Antti Karttunen, Feb 20 2015: (Start)
Among the terms a(1) .. a(8192), 1 occurs 4095 times, 2 occurs 1024 times, 3 occurs 2048 times and 4 occurs 1025 times. No larger numbers can ever occur.
That these are the first differences of not just A091067 and A255068, but also of A060833 follows from N. Sato's Feb 12 2013 comment in the latter that "For n > 1, n is in the sequence (A060833) if and only if A038189(n-1) = 1."
Also length of runs in A236840 and A255070.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A060833, A088742, A091067, A106837, A236840, A255058, A255068, A255070.

(define (A106836 n) (if (= 1 n) 3 (- (A091067 n) (A091067 (- n 1)))))

a(1) = 3, and for n > 1: a(n) = A091067(n) - A091067(n-1). - Antti Karttunen, Feb 20 2015

Name edited by Antti Karttunen, Feb 20 2015

A091067 Numbers whose odd part is of the form 4k+3.

Original entry on oeis.org

3, 6, 7, 11, 12, 14, 15, 19, 22, 23, 24, 27, 28, 30, 31, 35, 38, 39, 43, 44, 46, 47, 48, 51, 54, 55, 56, 59, 60, 62, 63, 67, 70, 71, 75, 76, 78, 79, 83, 86, 87, 88, 91, 92, 94, 95, 96, 99, 102, 103, 107, 108, 110, 111, 112, 115, 118, 119, 120, 123, 124, 126, 127, 131

Offset: 1

Ralf Stephan, Feb 22 2004

Either of form 2*a(m) or 4k+3, k >= 0, 0 < m < n.
A000265(a(n)) is an element of A004767.
a(n) such that A038189(a(n)) = 1.
Numbers n such that Kronecker(-n, m) = Kronecker(m, n) for all m. - Michael Somos, Sep 22 2005
From Antti Karttunen, Feb 20-21 2015: (Start)
Gives all n for which A005811(n) - A005811(n-1) = -1, from which follows that a(n) = the least k such that A255070(k) = n.
Gives the positions of even terms in A003602. (End)
Indices of negative terms in A164677. - M. F. Hasler, Aug 06 2015
Indices of the 0's in A014577. - Gabriele Fici, Jun 02 2016
Also indices of -1 in A034947. - Jianing Song, Apr 24 2021
Conjecture: alternate definition of same sequence is that a(1)=3 and a(n) is the smallest number > a(n-1) so that no number that is the sum of at most 2 terms in this sequence is a power of 2. - J. Lowell, Jan 20 2024
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Aug 31 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, On three conjectures of P. Barry, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Kevin Ryde, Iterations of the Dragon Curve, see index TurnRight, with a(n) = TurnRight(n-1).

Essentially one less than A060833.
Characteristic function: A038189.
Complement of A091072.
First differences are in A106836 (from its second term onward).
Sequence A246590 gives the even terms.
Gives the positions of records (after zero) for A255070 (equally, the position of the first n there).
Cf. A106837 (gives n such that both n and n+1 are terms of this sequence).
Cf. A098502 (gives n such that both n and n+2 are, but n+1 is not in this sequence).
Cf. also A000265, A003602, A004767, A005811, A014707, A034947, A055975, A236840, A255068, A255327, A255330.

import Data.List (elemIndices)
a091067 n = a091067_list !! (n-1)
a091067_list = map (+ 1) $ elemIndices 1 a014707_list
-- Reinhard Zumkeller, Sep 28 2011
(Scheme, with Antti Karttunen's IntSeq-library, two versions)
(define A091067 (MATCHING-POS 1 1 (COMPOSE even? A003602)))
(define A091067 (NONZERO-POS 1 0 A038189))
;; Antti Karttunen, Feb 20 2015

Select[Range[150], Mod[# / 2^IntegerExponent[#, 2], 4] == 3 &] (* Amiram Eldar, Aug 31 2024 *)

for(n=1,200,if(((n/2^valuation(n,2)-1)/2)%2,print1(n",")))

{a(n) = local(m, c); if( n<1, 0, c=0; m=1; while( cMichael Somos, Sep 22 2005 */

is_A091067(n)=bittest(n,valuation(n,2)+1) \\ M. F. Hasler, Aug 06 2015

a(n) = my(t=1); n<<=1; forstep(i=logint(n,2),0,-1, if(bittest(n,i)==t, n++;t=!t)); n; \\ Kevin Ryde, Mar 21 2021

a(n) = A060833(n+1) - 1. [See N. Sato's Feb 12 2013 comment in A060833.]
Other identities. For all n >= 1 it holds that:
A014707(a(n) + 1) = 1. - Reinhard Zumkeller, Sep 28 2011
A055975(a(n)) < 0. - Reinhard Zumkeller, Apr 28 2012
From Antti Karttunen, Feb 20-21 2015: (Start)
a(n) = A246590(n)/2.
A255070(a(n)) = n, or equally, A236840(a(n)) = 2n.
a(n) = 1 + A255068(n-1). (End)

A255327 a(n) = 0 if n is in the infinite trunk of "number-of-runs beanstalk" (one of the terms of A255056), otherwise number of nodes (including leaves and the node n itself) in that finite subtree of the beanstalk.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 5, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 0, 1, 5, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 2, 1, 10, 1, 0, 1, 8, 1, 0, 1, 3, 1, 2, 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 0, 1, 5, 1, 0, 1, 2, 1, 0

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

The edge-relation between nodes is given by A236840(child) = parent. a(n) = 1 + the size of transitive closure of all children emanating from the parent at n. For any n in A255056 this would be infinite, thus such n are marked with zeros.

Odd numbers are leaves, as there are no such k that A236840(k) were odd, thus a(2n+1) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..8590

Cf. A091067, A236840, A255056, A255068, A255328, A255329, A255330, A255339.

Cf. also A213727, A227643.

a(2n+1) = 1, and for even numbers 2n, if A255339(n) = 1, then a(2n) = 0, otherwise, a(2n) = 1 + sum_{k = A091067(n) .. A255068(n)} a(k).

A255330 a(n) = total number of nodes in the finite subtrees branching from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 7, 0, 3, 1, 0, 5, 2, 6, 0, 6, 0, 3, 1, 0, 5, 2, 12, 0, 2, 5, 0, 4, 2, 6, 0, 6, 0, 3, 1, 0, 5, 2, 12, 0, 2, 7, 1, 12, 4, 0, 2, 5, 0, 4, 2, 12, 0, 2, 5, 0, 4, 2, 6, 0, 6, 0, 3, 1, 0, 5, 2, 12, 0, 2, 7, 1, 12, 4, 0, 2, 7, 1, 10, 17, 0, 0, 1, 11, 4, 0, 2, 5, 0, 4, 2, 12, 0, 2, 7, 1, 12, 4, 0, 2, 5, 0, 4, 2, 12, 0, 2, 5, 0, 4, 2, 6, 0, 6, 0, 3, 1, 0, 5

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

A255058 gives the number of branches (children) of the node n in the trunk, of which one is the next node of the infinite trunk itself. Thus, if A255058(n) = 1, then a(n) = 0.

			The edge-relation between nodes is given by A236840(child) = parent. Odd numbers are leaves, as there are no such k that A236840(k) were odd.
The node 11 in the infinite trunk is A255056(11) = 30. Apart from 32 [we have A236840(32) = 30] which is the next node (node 12) in the infinite trunk, it has a single leaf-child 31 [A236840(31) = 30] at the "left side" (less than 32), and a leaf-child 33 [A236840(33) = 30] (more than 32) at the "right side", and also at that side, a subtree of three nodes 34 <- 38 <- 43 [we have A236840(43) = 38, A236840(38) = 34 and A236840(34) = 30], thus in total there are 1+1+3 = 5 nodes in finite branches emanating from the node 11 of the infinite trunk, and a(11) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..8590

Partial sums: A255333.

Cf. A091067, A236840, A255056, A255058, A255068, A255327, A255328, A255329, A255331.

(define (A255330 n) (if (zero? n) 1 (let ((k (A255057 n))) (add A255327 (A091067 k) (A255068 k)))))
;; Other code as in A255327.

a(0) = 1; a(n) = sum_{k = A091067(A255057(n)) .. A255068(A255057(n))} A255327(k).

a(n) = A255328(n) + A255329(n).

A255329 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 4, 1, 0, 0, 0, 0, 0, 1, 0, 4, 1, 0, 0, 1, 0, 0, 4, 1, 0, 0, 0, 0, 0, 1, 0, 4, 1, 0, 0, 1, 7, 1, 0, 3, 0, 1, 0, 0, 4, 1, 0, 0, 1, 0, 0, 4, 1, 0, 0, 0, 0, 0, 1, 0, 4, 1, 0, 0, 1, 7, 1, 0, 3, 0, 1, 7, 1, 0, 16, 0, 0, 1, 0, 3, 0, 1, 0, 0, 4, 1, 0, 0, 1, 7, 1, 0, 3, 0, 1, 0, 0, 4, 1, 0, 0, 1, 0, 0, 4, 1, 0, 0, 0, 0, 0, 1, 0, 4

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

			See example in A255330. Here we count only the nodes at the right side, thus a(11) = 1+3 = 4.

Antti Karttunen, Table of n, a(n) for n = 0..8590

Cf. A255056, A255057, A255327, A255328, A255330, A255331.

(define (A255329 n) (add A255327 (A255056 (+ n 1)) (A255068 (A255057 n))))
;; Other code as in A255327.

a(n) = sum_{k = A255056(n+1) .. A255068(A255057(n))} A255327(k).

a(n) = A255330(n) - A255328(n).

A255070 (1/2)*(n minus number of runs in the binary expansion of n): a(n) = (n - A005811(n)) / 2 = A236840(n)/2.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 9, 10, 11, 11, 11, 12, 13, 13, 14, 15, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 18, 19, 20, 20, 21, 22, 23, 23, 23, 24, 24, 24, 25, 26, 27, 27, 27, 28, 29, 29, 30, 31, 31, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 35

Offset: 0

Antti Karttunen, Feb 14 2015

Antti Karttunen, Table of n, a(n) for n = 0..8192
Helmut Prodinger and Friedrich J. Urbanek, Infinite 0-1-Sequences Without Long Adjacent Identical Blocks, Discrete Mathematics, Volume 28, Number 3, 1979, pages 277-289. Also first author's copy. See theorem 3.6, n_1(k) = a(k).

Least inverse: A091067 (also the positions of records).
Greatest inverse: A255068.
Run lengths: A106836.
Cf. A005811, A060833, A236840, A255054, A255056, A255072, A269363, A269367.

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

Scheme
```
(define (A255070 n) (/ (A236840 n) 2))
```

a(n) = A236840(n) / 2 = (n - A005811(n)) / 2.
Other identities:
a(A091067(n)) = n for all n >= 1.
a(A255068(n)) = n for all n >= 0.
a(A269363(n)) = A269367(n). - Antti Karttunen, Aug 12 2019

A322015 If A003188(n+1) < A003188(n), then a(n) = n+1, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 3, 0, 0, 6, 7, 0, 0, 0, 11, 12, 0, 14, 15, 0, 0, 0, 19, 0, 0, 22, 23, 24, 0, 0, 27, 28, 0, 30, 31, 0, 0, 0, 35, 0, 0, 38, 39, 0, 0, 0, 43, 44, 0, 46, 47, 48, 0, 0, 51, 0, 0, 54, 55, 56, 0, 0, 59, 60, 0, 62, 63, 0, 0, 0, 67, 0, 0, 70, 71, 0, 0, 0, 75, 76, 0, 78, 79, 0, 0, 0, 83, 0, 0, 86, 87, 88, 0, 0, 91, 92, 0, 94, 95, 96

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A322016, A322017.

Cf. also A091067, A255068.

With[{nn = 96}, ReplacePart[ConstantArray[0, nn], Map[# -> # &, 1 + Position[Partition[Array[BitXor[#, Floor[#/2]] &, nn], 2, 1], ?(First@ # > Last@ # &), {1}, Heads -> False][[All, 1]] ]]] (* _Michael De Vlieger, Nov 25 2018 *)

A003188(n) = bitxor(n, n>>1);
A322015(n) = if(A003188(1+n)<A003188(n),1+n,0);

A341522 a(n) = A156552(3*A005940(1+n)).

Original entry on oeis.org

2, 5, 6, 11, 10, 13, 14, 23, 18, 21, 22, 27, 26, 29, 30, 47, 34, 37, 38, 43, 42, 45, 46, 55, 50, 53, 54, 59, 58, 61, 62, 95, 66, 69, 70, 75, 74, 77, 78, 87, 82, 85, 86, 91, 90, 93, 94, 111, 98, 101, 102, 107, 106, 109, 110, 119, 114, 117, 118, 123, 122, 125, 126, 191, 130, 133, 134, 139, 138, 141, 142, 151, 146, 149

Offset: 0

Antti Karttunen, Feb 15 2021

nonn

Because the least significant 0-bit in A156552-code of any nonzero multiple of 3 is always alone (has 1-bit immediately to its left), it follows that A255068 (= A091067(n+1) - 1) gives these same terms in the ascending order.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for sequences related to binary expansion of n

Row/column 2 of A341520. Permutation of A255068.

Cf. A005940, A007814, A156552, A086799, A014707 (characteristic function).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341522(n) = A156552(3*A005940(1+n));

a(n) = A156552(3*A005940(1+n)).
From Antti Karttunen, Feb 23 2021: (Start)
a(n) = 1 + n + A086799(1+n). - [Conjectured by LODA-miner, and easily seen to be correct]
a(n) = 1+ 2*n + 2^A007814(1+n). - [As the above can be rewritten to this]
(End)

cp's OEIS Frontend

A106836 First differences of A060833 and (from a(2) onward) also of A091067 and A255068.

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A091067 Numbers whose odd part is of the form 4k+3.

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A255327 a(n) = 0 if n is in the infinite trunk of "number-of-runs beanstalk" (one of the terms of A255056), otherwise number of nodes (including leaves and the node n itself) in that finite subtree of the beanstalk.

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A255330 a(n) = total number of nodes in the finite subtrees branching from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

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A255329 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

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A255070 (1/2)*(n minus number of runs in the binary expansion of n): a(n) = (n - A005811(n)) / 2 = A236840(n)/2.

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A322015 If A003188(n+1) < A003188(n), then a(n) = n+1, otherwise a(n) = 0.

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A341522 a(n) = A156552(3*A005940(1+n)).

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