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.