Colour, read and jump sequence
Modus
operandi:
Write
the seq. N of the natural numbers:
N = 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 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44 45 ...
Rule one: start with ‘n’=1
Rule two: at step ‘n’, the
yellow integer ‘n’ has to jump over k
integers to the right (k is given by
the integer on the right-side of ‘n’)
Rule three:
make ‘n’=‘n’+1 then go back to Rule one
and apply again to the re-ordered set of Naturals.
[The
first column on the left (in blue) shows the step of the process]
1
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 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45 ...
2 2
3 1
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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
3 3
1 4 2 5 6
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
4 1 3 4 2 5
6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45 ...
5 1
3 2 5 4
6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45 ...
6 1
3 2 4 6 7
8 5 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44 45 ...
7 1
3 2 4 7 8
5 9 10 11 12 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
8 1
3 2 4 8 5 9 10
11 12 6 13 7 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
9 1
3 2 4
5 9 10 11 12 8 6 13 7
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 ...
10 1 3
2 4 5 10 11 12 8 6 13 7
14 15 16 9 17 18 19 20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
11 1 3
2 4 5 11 12 8 6 13 7
14 15 16 9 17 10 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
12 1 3
2 4 5 12 8 6 13 7
14 15 16 9 17 10 18 11 19 20 21 22 23 24 25 26 27 28 29 30 31
32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
13 1 3
2 4 5
8 6 13 7 14 15
16 9 12 17 10 18 11 19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
14 1 3
2 4 5 8 6 7 14 15 16 9 12 17 13 10 18 11 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
34 35 36 37 38 39 40 41 42 43 44 45 ...
15 1 3
2 4 5 8 6 7 15 16 9 12 17 13 10 18 11 19 20 21 22 23 24 14 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
16 1 3
2 4 5 8 6 7 16 9 12 17 13 10 18 11 19 20 21 22 23 24 14 25 15 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
17 1 3
2 4 5 8 6
7 9 12 17 13 10 18 11 19 20 16 21 22 23 24 14 25 15 26 27 28 29 30 31
32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
18 1 3
2 4 5 8 6
7 9 12 13 10 18 11 19 20 16 21 22 23
24 14 25 17
15 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
19 1 3
2 4 5 8 6
7 9 12 13 10 11 19 20 16 21 22 23 24 14
25 17 18
15 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ...
20 1 3 2 4
5 8 6
7 9 12 13 10 11 20 16 21 22 23 24 14 25
17 18 15 26 27 28 29 30 31 32 33 34 19 35 36 37 38 39 40 41 42 43 44 45 ...
21 1 3
2 4 5 8 6
7 9 12 13 10 11 16 21 22 23 24 14 25 17 18
15 26 27 28 29 30 31 20 32 33 34 19 35 36 37 38 39 40 41 42 43 44 45 ...
22 1 3
2 4 5 8 6
7 9 12 13 10 11 16 22 23 24 14 25 17 18 15
26 27 28 29 30 31 20 32 33 34 19 35 36 37 21 38 39 40 41 42 43 44 45 ...
...
We see that at step ‘1’, the yellow integer ‘1’ has to jump over 2
integers to the right (ending between 3 and 4)
at
step ‘2’, the yellow integer ‘2’ has to jump over 3 integers to the right
(ending between 4 and 5)
at
step ‘3’, the yellow integer ‘3’ has to jump over 1 integer to the right
(ending between 1 and 4); etc.
Emerging
sequence : 1, 3, 2, 4, 5, 8, 6, 7, 9, 12, 13, 10,
11, 16, 14, 17, 18, 15, 20, 24, 19, ...
Could
someone please compute a few more terms of this (very slow emerging) sequence?
What
are the integers, in this sequence, which keep the same rank as in N? (see the answer from Jean-Marc Falcoz, below)
Best,
É.
---
Update, July 14th, 2009 --> 11:40, Brussels time:
Pr. Dr. Alois Heinz helped
again (see here), corrected and extended the sequence:
> The first 666 items:
1,
3, 2, 4, 5, 8, 6, 7, 9, 12, 13, 10, 11, 16, 14, 17, 18, 15, 20, 24, 19, 25, 21,
22, 23, 31, 26, 34, 27, 33, 28, 37, 29, 38, 36, 30, 39, 32, 35, 46, 44, 47, 40,
48, 41, 42, 43, 49, 50, 53, 45, 58, 51, 59, 61, 52, 67, 54, 68, 55, 69, 56, 70,
57, 60, 73, 62, 63, 64, 65, 66, 80, 90, 71, 72, 74, 91, 94, 75, 76, 77, 78, 79,
93, 81, 95, 82, 83, 96, 102, 84, 97, 85, 86, 87, 107, 88, 108, 89, 109, 110,
106, 92, 98, 122, 118, 99, 100, 101, 127, 103, 104, 129, 105, 121, 130, 131,
132, 135, 111, 128, 112, 137, 113, 114, 138, 134, 115, 116, 136, 140, 117, 141,
119, 120, 139, 123, 124, 125, 126, 150, 142, 143, 144, 147, 146, 133, 164, 166,
167, 170, 171, 165, 145, 148, 149, 151, 172, 184, 152, 185, 153, 154, 186, 174,
187, 155, 156, 176, 188, 157, 189, 158, 159, 190, 160, 182, 161, 193, 162, 194,
163, 200, 201, 168, 202, 169, 203, 173, 175, 177, 178, 179, 207, 180, 204, 218,
210, 181, 211, 183, 213, 191, 192, 195, 225, 196, 197, 227, 247, 198, 199, 252,
233, 253, 240, 235, 256, 205, 257, 206, 208, 259, 209, 212, 261, 214, 262, 215,
263, 216, 264, 217, 265, 260, 219, 220, 221, 258, 222, 223, 224, 226, 228, 266,
229, 230, 231, 280, 232, 281, 234, 267, 236, 268, 237, 238, 239, 291, 241, 242,
294, 243, 244, 277, 297, 245, 298, 279, 299, 246, 300, 248, 249, 250, 251, 254,
255, 304, 301, 326, 327, 333, 269, 270, 335, 271, 272, 273, 337, 274, 275, 276,
278, 334, 282, 283, 284, 349, 285, 350, 286, 287, 288, 353, 289, 354, 290, 292,
293, 359, 336, 295, 296, 362, 363, 345, 342, 343, 302, 303, 370, 346, 371, 305,
306, 373, 307, 374, 308, 309, 377, 310, 311, 312, 380, 313, 381, 314, 382, 315,
383, 316, 317, 364, 318, 365, 319, 387, 369, 320, 372, 367, 321, 390, 322, 391,
323, 392, 324, 393, 325, 396, 328, 397, 329, 330, 398, 331, 399, 332, 400, 388,
384, 404, 405, 338, 406, 386, 339, 407, 340, 341, 412, 413, 344, 347, 401, 394,
348, 351, 352, 355, 356, 357, 358, 403, 360, 361, 366, 368, 453, 454, 455, 456,
457, 437, 436, 459, 460, 375, 376, 463, 378, 379, 473, 385, 439, 474, 475, 441,
389, 477, 478, 479, 450, 395, 483, 484, 458, 488, 402, 472, 408, 409, 410, 503,
411, 504, 414, 476, 415, 487, 509, 416, 510, 417, 511, 418, 419, 512, 480, 513,
420, 421, 422, 515, 423, 516, 424, 497, 517, 485, 425, 426, 500, 427, 428, 429,
430, 521, 491, 431, 522, 432, 433, 434, 435, 438, 440, 526, 505, 442, 443, 506,
444, 445, 446, 447, 448, 449, 451, 514, 452, 536, 537, 523, 538, 518, 519, 520,
525, 461, 462, 524, 464, 465, 466, 467, 468, 554, 469, 555, 527, 470, 471, 528,
529, 534, 567, 535, 568, 544, 550, 574, 481, 575, 540, 482, 581, 486, 582, 583,
584, 489, 552, 490, 588, 553, 589, 492, 493, 494, 495, 496, 498, 559, 601, 499,
561, 604, 501, 502, 507, 508, 620, 623, 624, 530, 644, 531, 532, 533, 616, 617,
606, 609, 656, 611, 539, 613, 541, 542, 543, 619, 545, 546, 547, 622, 548, 549,
675, 551, 556, 557, 686, 637, 687, 558, 688, 654, 689, 560, 640, 562, 563, 694,
564, 695, 565, 566, 696, 646, 697, 666, 649, 699, 569, 570, 571, 572, 701, 573,
702, 658, 576, 577, 578, 579, 677, 580, 706, 667, 707, 708, 709, 710, 585, 586,
587, 713, 590, 714, 591, 715, 592, 593, 594, 595, 720, 596, 597, 598, 723, 599,
600, 602, 603, 727, 728, 605, 729, 730, 607, 731, 608, 732, 733, 610, 698, 736,
705, 612, 737, 614, 700, ...
... thank you, Alois !
Grüsse,
É.
__________
(the full b-file with 10000 terms is there)
__________
Update February 21st, 2011 (in French)
I’ve just received this from Jean-Marc
Falcoz:
J’ai regardé de plus près une de tes anciennes suites « ColourReadJump ».
Tu posais la question de savoir quels sont les entiers qui gardent leur place dans N [donc f(n)=n]
Je suis allé jusqu’à 25000, et les seuls qui vérifient cette condition sont :
{1, 4, 5, 9, 142, 143, 144, 146}
J’ai fait 3 graphiques de cette suite :
a) jusqu’à 150 : on voit bien les 8 points tels que f(n)=n, car ils sont sur la diagonale pointillée.
b) jusqu’à 1500 : on voit émerger 3 ou 4 « lignes ».... curieux !
c) jusqu’à 15000 : les 4 lignes se précisent...
C’est vraiment étrange, ce comportement.
Intriguing graph of the sequence, indeed: four lines!
Merci beaucoup, Jean-Marc!
Best,
É.
(back to main page, là
)