Tag: relation

This is inspired by a question math.SE, where an infinite sequence of by different pairs natural numbers $ a_1 = 1, a_2, a_3, … $ It has been defined as follows:
$ a_n $ is the smallest number such that $ s_n: = sqrt {a_n + sqrt {a_ {n-1} + sqrt { cdots + sqrt {a_1}}}} $ it is a whole
It turns out that this sequence, which by the way is not yet in the OEIS, is in fact a permutation of $ mathbb N $. In addition, the images show that both $ (a_n) $Y $ (s_n) $ They exhibit an interesting self-similarity, with two alternate structures and the relationship that converges rapidly towards, as it seems, $ 3 + 2 sqrt2 = ( sqrt2 + 1) ^ 2 approx 5.828427 $ (see this other answer). Next I have shown the first $ 632 $ entries in a way that makes it easier to see what numbers generate what is perceived in the images as lines.

one,
3, 2,
7, 6,
13, 5,
22, 4,
33, 10, 12, 21, 11,
32, 19, 20,
31, 30,

43, 9, 45, 18, 44, 29,
58, 8, 60, 17, 59, 28,
75, 16, 76, 27,
94, 15, 95, 26,
115, 14, 116, 25,
138, 24,
163, 23,
190, 35, 42, 57, 41, 74, 40, 93, 39, 114, 38, 137, 37, 162, 36,
189, 50, 56, 73, 55, 92, 54, 113, 53, 136, 52, 161, 51,
188, 67, 72, 91, 71, 112, 70, 135, 69, 160, 68,
187, 86, 90, 111, 89, 134, 88, 159, 87,
186, 107, 110, 133, 109, 158, 108,
185, 130, 132, 157, 131,
184, 155, 156,
183, 182,

211, 34, 218, 49, 217, 66, 216, 85, 215, 106, 214, 129, 213, 154, 212, 181,
242, 48, 248, 65, 247, 84, 246, 105, 245, 128, 244, 153, 243, 180,
275, 47, 281, 64, 280, 83, 279, 104, 278, 127, 277, 152, 276, 179,
310, 46, 316, 63, 315, 82, 314, 103, 313, 126, 312, 151, 311, 178,
347, 62, 352, 81, 351, 102, 350, 125, 349, 150, 348, 177,
386, 61, 391, 80, 390, 101, 389, 124, 388, 149, 387, 176,
427, 79, 431, 100, 430, 123, 429, 148, 428, 175,
470, 78, 474, 99, 473, 122, 472, 147, 471, 174,
515, 77, 519, 98, 518, 121, 517, 146, 516, 173,
562, 97, 565, 120, 564, 145, 563, 172,
611, 96, 614, 119, 613, 144, 612, 171,
662, 118, 664, 143, 663, 170,
715, 117, 717, 142, 716, 169,
770, 141, 771, 168,
827, 140, 828, 167,
886, 139, 887, 166,
947, 165,
1010, 164,
1075, 192, 210, 241, 209, 274, 208, 309, 207, 346, 206, 385, 205, 426, 204, 469, 203, 514, 202, 561, 201, 610, 200, 661, 199, 714, 198, 769, 197, 826, 196, 885, 195, 946, 194, 1009, 193,
1074, 223, 240, 273, 239, 308, 238, 345, 237, 384, 236, 425, 235, 468, 234, 513, 233, 560, 232, 609, 231, 660, 230, 713, 229, 768, 228, 825, 227, 884, 226, 945, 225, 1008, 224,
1073, 256, 272, 307, 271, 344, 270, 383, 269, 424, 268, 467, 267, 512, 266, 559, 265, 608, 264, 659, 263, 712, 262, 767, 261, 824, 260, 883, 259, 944, 258, 1007, 257,
1072, 291, 306, 343, 305, 382, ​​304, 423, 303, 466, 302, 511, 301, 558, 300, 607, 299, 658, 298, 711, 297, 766, 296, 823, 295, 882, 294, 943, 293, 1006, 292,
1071, 328, 342, 381, 341, 422, 340, 465, 339, 510, 338, 557, 337, 606, 336, 657, 335, 710, 334, 765, 333, 822, 332, 881, 331, 942, 330, 1005, 329,
1070, 367, 380, 421, 379, 464, 378, 509, 377, 556, 376, 605, 375, 656, 374, 709, 373, 764, 372, 821, 371, 880, 370, 941, 369, 1004, 368,
1069, 408, 420, 463, 419, 508, 418, 555, 417, 604, 416, 655, 415, 708, 463, 413, 820, 412, 879, 411, 940, 410, 1003, 409,
1068, 451, 462, 507, 461, 554, 460, 603, 459, 654, 458, 707, 457, 762, 456, 819, 455, 878, 454, 939, 453, 1002, 452,
1067, 496, 506, 553, 505, 602, 504, 653, 503, 706, 502, 761, 501, 818, 500, 877, 499, 938, 498, 1001, 497,
1066, 543, 552, 601, 551, 652, 550, 705, 549, 760, 548, 817, 547, 876, 546, 937, 545, 1000, 544,
1065, 592, 600, 651, 599, 704, 598, 759, 597, 816, 596, 875, 595, 936, 594, 999, 593,
1064, 643, 650, 703, 649, 758, 648, 815, 647, 874, 646, 935, 645, 998, 644,
1063, 696, 702, 757, 701, 814, 700, 873, 699, 934, 698, 997, 697,
1062, 751, 756, 813, 755, 872, 754, 933, 753, 996, 752,
1061, 808, 812, 871, 811, 932, 810, 995, 809,
1060, 867, 870, 931, 869, 994, 868,
1059, 928, 930, 993, 929,
1058, 991, 992,
1057, 1056,

1123, 191, ....  

Once the data is sorted like this, the patterns seem quite predictable. However, every other block (for example, the penultimate one, which starts with $ 211 = a_ {113} $) has "paragraphs" of lengths $ 2 $ or $ 3 $, except possibly the first. Now you can see by construction that from a block to the next (envelope), the sequence 2-3 of the block is generated by the previous one in a similar way to the "rabbit sequence", also known as "Fibonacci". word "https://oeis.org/A005614, by the laws (essentially) $ 3 to22, 2 to323 $ plus the boundary conditions that are much harder to predict …
So this partly explains self-similarity. But:

How can you prove that the asymptotic relationship is $ 3 + 2 sqrt2 $?

Each block consists of a group of monotonous "horizontal" subsequences and a group of monotonous "vertical" subsequences. For each other block, they come in roughly "L-shaped" pairs. The numbers of "L shapes" per block are clearly distinguishable, p. Ex. $ 8 $ of them in the range of $ n = 49, dots, $ 112 (starting after $ a_ {48} = 190 $) Y $ 19 $ for $ n = 270, dots, $ 630 (starting after $ a_ {269} = $ 1075, the beginning of the last block). Those numbers $ (c_j) = 3,8,19,46, points $ it seems that they form the sequence of Fibonacci types https://oeis.org/A078343 with $$ c_j = frac14 Bigl[(3 sqrt{2} – 2) (1 + sqrt{2})^j – (3 sqrt{2}+2) (1 – sqrt{2})^jBigr], $$ which is another indication in favor of the conjectured relationship, but I'm not sure if the recursion $ c_j = 2c_ {j-1} + c_ {j-2} $ It can be shown by induction.

You can also take a look at https://codegolf.stackexchange.com/a/145234/14614, which shows the differences $ a_ {n + 1} -a_n $, and in the image of the reverse map. $ a_n mapsto n $ cited below one of the comments. (Note that the isolated point in $ a_n = $ 191 corresponds to $ n = $ 632, which is just where my previous table stops.)
Both show a lot of beauty, but they also show that self-similarity is somewhat less strict than for the fractal sequences mentioned here.