A103621 -id:A103621 - cp's OEIS frontend

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

0, 3, 6, 5, 4, 15, 10, 9, 8, 11, 14, 13, 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30, 61, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 126, 93, 76, 71, 66, 65, 64, 67, 70, 69

Offset: 0

Philippe Deléham, Feb 13 2005

All terms are distinct, but certain terms (see A102371) are missing. But see A103122.

Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.

			........0
........1
.......10
.......11
......100
......101
......110
......111
.....1000
.........
The upward-sloping diagonals are:
0
11
110
101
100
1111
1010
.......
giving 0, 3, 6, 5, 4, 15, 10, ...
The sequence has a natural decomposition into blocks (see the paper): 0; 3; 6, 5, 4; 15, 10, 9, 8, 11, 14, 13; 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30; 61, ...
Reading the array of binary numbers along diagonals with slope 1 gives this sequence, slope 2 gives A105085, slope 0 gives A001477 and slope -1 gives A105033.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].
Index entries for sequences related to binary expansion of n

Related sequences (1): A103542 (binary version), A102371 (complement), A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, A103582, A103583.
Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 (base 10 version).
Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, A105026, A105027, A105028.
Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, A105035, A105108.
Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, A104490, A104853, A104893, A104894, A105085.

a102370 n = a102370_list !! n
a102370_list = 0 : map (a105027 . toInteger) a062289_list
-- Reinhard Zumkeller, Jul 21 2012

A102370:=proc(n) local t1,l; t1:=n; for l from 1 to n do if n+l mod 2^l = 0 then t1:=t1+2^l; fi; od: t1; end;

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ f[n] + n, {n, 0, 71}] (* Robert G. Wilson v, Mar 21 2005 *)

A102370(n)=n-1+sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k)) \\ Benoit Cloitre, Mar 20 2005

{a(n) = if( n<1, 0, sum( k=0, length( binary( n)), bitand( n + k, 2^k)))} /* Michael Somos, Mar 26 2012 */

def a(n): return 0 if n<1 else sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)]) # Indranil Ghosh, May 03 2017

a(n) = n + Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k. (Cf. A103185.) In particular, a(n) >= n. - N. J. A. Sloane, Mar 18 2005

a(n) = A105027(A062289(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

More terms from Benoit Cloitre, Mar 20 2005

A103192 Trajectory of 1 under repeated application of the function n -> A102370(n).

Original entry on oeis.org

1, 3, 5, 15, 17, 19, 21, 31, 33, 35, 37, 47, 49, 51, 53, 63, 65, 67, 69, 79, 81, 83, 85, 95, 97, 99, 101, 111, 113, 115, 117, 127, 129, 131, 133, 143, 145, 147, 149, 159, 161, 163, 165, 175, 177, 179, 181, 191, 193, 195, 197, 207, 209, 211, 213, 223, 225, 227, 229, 239, 241

Offset: 1

N. J. A. Sloane, David Applegate, Benoit Cloitre and Philippe Deléham, Mar 25 2005

Agrees with A103127 for the first 511 terms, but then diverges. If a(n) is the present sequence and b(n) is A103127 we have:
.n...a(n)..b(n)..difference
.....................
509, 2033, 2033, 0
510, 2035, 2035, 0
511, 2037, 2037, 0
512, 4095, 2047, 2048
513, 4097, 2049, 2048
514, 4099, 2051, 2048
515, 4101, 2053, 2048
516, 4111, 2063, 2048
.....................
The sequence may be computed as follows (from Philippe Deléham, May 08 2005).
Start with -1, 1. Then add powers of 2 whose exponent n is not in A034797: 1, 3, 11, 2059, 2^2059 + 2059, ... This gives
Step 0: -1, 1
Step 1: add 2^2 = 4, getting 3, 5 and thus -1, 1, 3, 5.
Step 2: add 2^4 = 16, getting 15, 17, 19, 21 and thus -1, 1, 3, 5, 15, 17, 19, 21
Step 3: add 2^5 = 32, getting 31, 33, 35, 37, 47, 49, 51, 53 and thus -1, 1, 3, 5, 15, 17, 19, 21, 31, 33, 35, 37, 47, 49, 51, 53, etc.
The jump 2037 --> 4095 for n = 510 -> 511 is explained by the fact that we pass directly from 2^10 to 2^12 since 11 belongs to A034797.
The trajectories of 2 (A103747) and 7 (A103621) may surely be obtained in a similar way.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

a103192 n = a103192_list !! (n-1)
a103192_list = iterate (fromInteger . a102370) 1
-- Reinhard Zumkeller, Jul 21 2012

A103747 Trajectory of 2 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 254, 258, 262, 266, 270, 274, 278, 282, 286, 290, 294, 298, 302, 306, 310, 314, 382, 386, 390, 394, 398, 402, 406, 410, 414, 418, 422

Offset: 1

Benoit Cloitre and David Applegate, Mar 25 2005

Although it initially appears that a(n)-8n is the 16-periodic sequence {-2,-6,-10,-14,-18,-22,-26,-30,-34,-38,-42,-46,-50,-54,6,2}, this pattern eventually breaks down. However, the first divergence occurs beyond the first 400 million terms.

(a(n)) agrees with the 16-periodic sequence up to a(2^67-1) = 2^70 - 70, but then diverges with a(2^67) = 2^71 - 2. - Charlie Neder, Feb 07 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps], preprint, 2005.
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

Cf. A102370, A132417.

Trajectories of other numbers: A103192 (1), A103621 (7), A158953 (12), A159887 (29).

a103747 n = a103747_list !! (n-1)
a103747_list = iterate (fromInteger . a102370) 2
-- Reinhard Zumkeller, Jul 21 2012

Edited by Peter Munn, Jan 13 2024

A158953 Trajectory of 12 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

12, 28, 44, 60, 76, 92, 108, 124, 140, 156, 172, 188, 204, 220, 236, 252, 268, 284, 300, 316, 332, 348, 364, 380, 396, 412, 428, 444, 460, 476, 492, 508, 524, 540, 556, 572, 588, 604, 620, 636, 652, 668, 684, 700, 716, 732, 748, 764, 780, 796, 812, 828, 844

Offset: 1

Philippe Deléham, Apr 01 2009

Coincides with A098502 for at least 1400 terms. - R. J. Mathar, Apr 16 2009

Agrees with A098502 for the first 65535 terms. A098502(65535) = a(65535) = 1048556 = 2^20 - 20. A098502(65536) = 1048572 = 2^20 - 4; a(65536) = 2097148 = 2^21 - 4. - Philippe Deléham, Jan 05 2023

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].
Index entries for sequences which agree for a long time but are different

Cf. A098502, A102370.

Trajectories of other numbers: A103192 (1), A103747 (2), A103621 (7), A159887 (29).

A159887 Trajectory of 29 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

29, 39, 41, 43, 45, 55, 57, 59, 93, 103, 105, 107, 109, 119, 121, 251, 285, 295, 297, 299, 301, 311, 313, 315, 349, 359, 361, 363, 365, 375, 377, 507, 541, 551, 553, 555, 557, 567, 569, 571, 605, 615, 617, 619, 621, 631, 633, 763, 797, 807, 809, 811, 813, 823, 825

Offset: 1

Philippe Deléham, Apr 25 2009

Not the same as A159888: see the comments in A159888.

The divergence from A159888 follows from Theorem 3.1 in the Applegate, Cloitre, Deléham and Sloane link: in general, the first differences of an A102370 trajectory cannot be a cycle. - Peter Munn, Jan 14 2024

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

Cf. A102370, A159888.

Trajectories of other numbers: A103192 (1), A103747 (2), A103621 (7), A158953 (12).

Missing term 617 inserted by Georg Fischer, Nov 28 2023

A158938 Numbers congruent to {-3,7,9,11,13,23,25,27} mod 64.

Original entry on oeis.org

-3, 7, 9, 11, 13, 23, 25, 27, 61, 71, 73, 75, 77, 87, 89, 91, 125, 135, 137, 139, 141, 151, 153, 155, 189, 199, 201, 203, 205, 215, 217, 219, 253, 263, 265, 267, 269, 279, 281, 283, 317, 327, 329, 331, 333, 343, 345, 347, 381, 391, 393, 395, 397, 407, 409, 411

Offset: 0

Philippe Deléham, Mar 31 2009

When will this first differ from A103621?

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A103621.

Select[Range[-5,500],MemberQ[{7,9,11,13,23,25,27,61},Mod[#,64]]&] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{-3,7,9,11,13,23,25,27,61},60] (* Harvey P. Dale, Jul 26 2021 *)

a(n) = a(n-8) + 64.
From Colin Barker, Oct 14 2019: (Start)
G.f.: -(3 - 10*x - 2*x^2 - 2*x^3 - 2*x^4 - 10*x^5 - 2*x^6 - 2*x^7 - 37*x^8) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = a(n-1) + a(n-8) - a(n-9) for n>8.
(End)

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A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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A103192 Trajectory of 1 under repeated application of the function n -> A102370(n).

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A103747 Trajectory of 2 under repeated application of the map n -> A102370(n).

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A158953 Trajectory of 12 under repeated application of the map n -> A102370(n).

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A159887 Trajectory of 29 under repeated application of the map n -> A102370(n).

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A158938 Numbers congruent to {-3,7,9,11,13,23,25,27} mod 64.

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