An integer sequence puzzle

[CraigD]

As I hinted, modest surmised a couple of days ago, and I believe Turtle’s suggesting, a step in how I generated the sequences in post #1 was to sum the digits of the members of another integer sequence. This step of the puzzle solved, we can rewrite the sequences (for consistency’s sake, their first 512 terms):

1, 5, 5, 17, 13, 37, 25, 65, 41, 101, 61, 145, 85, 197, 113, 257, 145, 325, 181, 401, 221, 485, 265, 577, 313, 677, 365, 785, 421, 901, 481, 1025, 545, 1157, 613, 1297, 685, 1445, 761, 1601, 841, 1765, 925, 1937, 1013, 2117, 1105, 2305, 1201, 2501, 1301, 2705, 1405, 2917, 1513, 3137, 1625, 3365, 1741, 3601, 1861, 3845, 1985, 4097, 2113, 4357, 2245, 4625, 2381, 4901, 2521, 5185, 2665, 5477, 2813, 5777, 2965, 6085, 3121, 6401, 3281, 6725, 3445, 7057, 3613, 7397, 3785, 7745, 3961, 8101, 4141, 8465, 4325, 8837, 4513, 9217, 4705, 9605, 4901, 10001, 5101, 10405, 5305, 10817, 5513, 11237, 5725, 11665, 5941, 12101, 6161, 12545, 6385, 12997, 6613, 13457, 6845, 13925, 7081, 14401, 7321, 14885, 7565, 15377, 7813, 15877, 8065, 16385, 8321, 16901, 8581, 17425, 8845, 17957, 9113, 18497, 9385, 19045, 9661, 19601, 9941, 20165, 10225, 20737, 10513, 21317, 10805, 21905, 11101, 22501, 11401, 23105, 11705, 23717, 12013, 24337, 12325, 24965, 12641, 25601, 12961, 26245, 13285, 26897, 13613, 27557, 13945, 28225, 14281, 28901, 14621, 29585, 14965, 30277, 15313, 30977, 15665, 31685, 16021, 32401, 16381, 33125, 16745, 33857, 17113, 34597, 17485, 35345, 17861, 36101, 18241, 36865, 18625, 37637, 19013, 38417, 19405, 39205, 19801, 40001, 20201, 40805, 20605, 41617, 21013, 42437, 21425, 43265, 21841, 44101, 22261, 44945, 22685, 45797, 23113, 46657, 23545, 47525, 23981, 48401, 24421, 49285, 24865, 50177, 25313, 51077, 25765, 51985, 26221, 52901, 26681, 53825, 27145, 54757, 27613, 55697, 28085, 56645, 28561, 57601, 29041, 58565, 29525, 59537, 30013, 60517, 30505, 61505, 31001, 62501, 31501, 63505, 32005, 64517, 32513, 65537, 33025, 66565, 33541, 67601, 34061, 68645, 34585, 69697, 35113, 70757, 35645, 71825, 36181, 72901, 36721, 73985, 37265, 75077, 37813, 76177, 38365, 77285, 38921, 78401, 39481, 79525, 40045, 80657, 40613, 81797, 41185, 82945, 41761, 84101, 42341, 85265, 42925, 86437, 43513, 87617, 44105, 88805, 44701, 90001, 45301, 91205, 45905, 92417, 46513, 93637, 47125, 94865, 47741, 96101, 48361, 97345, 48985, 98597, 49613, 99857, 50245, 101125, 50881, 102401, 51521, 103685, 52165, 104977, 52813, 106277, 53465, 107585, 54121, 108901, 54781, 110225, 55445, 111557, 56113, 112897, 56785, 114245, 57461, 115601, 58141, 116965, 58825, 118337, 59513, 119717, 60205, 121105, 60901, 122501, 61601, 123905, 62305, 125317, 63013, 126737, 63725, 128165, 64441, 129601, 65161, 131045, 65885, 132497, 66613, 133957, 67345, 135425, 68081, 136901, 68821, 138385, 69565, 139877, 70313, 141377, 71065, 142885, 71821, 144401, 72581, 145925, 73345, 147457, 74113, 148997, 74885, 150545, 75661, 152101, 76441, 153665, 77225, 155237, 78013, 156817, 78805, 158405, 79601, 160001, 80401, 161605, 81205, 163217, 82013, 164837, 82825, 166465, 83641, 168101, 84461, 169745, 85285, 171397, 86113, 173057, 86945, 174725, 87781, 176401, 88621, 178085, 89465, 179777, 90313, 181477, 91165, 183185, 92021, 184901, 92881, 186625, 93745, 188357, 94613, 190097, 95485, 191845, 96361, 193601, 97241, 195365, 98125, 197137, 99013, 198917, 99905, 200705, 100801, 202501, 101701, 204305, 102605, 206117, 103513, 207937, 104425, 209765, 105341, 211601, 106261, 213445, 107185, 215297, 108113, 217157, 109045, 219025, 109981, 220901, 110921, 222785, 111865, 224677, 112813, 226577, 113765, 228485, 114721, 230401, 115681, 232325, 116645, 234257, 117613, 236197, 118585, 238145, 119561, 240101, 120541, 242065, 121525, 244037, 122513, 246017, 123505, 248005, 124501, 250001, 125501, 252005, 126505, 254017, 127513, 256037, 128525, 258065, 129541, 260101, 130561, 262145

and

1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128, 65, 132, 67, 136, 69, 140, 71, 144, 73, 148, 75, 152, 77, 156, 79, 160, 81, 164, 83, 168, 85, 172, 87, 176, 89, 180, 91, 184, 93, 188, 95, 192, 97, 196, 99, 200, 101, 204, 103, 208, 105, 212, 107, 216, 109, 220, 111, 224, 113, 228, 115, 232, 117, 236, 119, 240, 121, 244, 123, 248, 125, 252, 127, 256, 129, 260, 131, 264, 133, 268, 135, 272, 137, 276, 139, 280, 141, 284, 143, 288, 145, 292, 147, 296, 149, 300, 151, 304, 153, 308, 155, 312, 157, 316, 159, 320, 161, 324, 163, 328, 165, 332, 167, 336, 169, 340, 171, 344, 173, 348, 175, 352, 177, 356, 179, 360, 181, 364, 183, 368, 185, 372, 187, 376, 189, 380, 191, 384, 193, 388, 195, 392, 197, 396, 199, 400, 201, 404, 203, 408, 205, 412, 207, 416, 209, 420, 211, 424, 213, 428, 215, 432, 217, 436, 219, 440, 221, 444, 223, 448, 225, 452, 227, 456, 229, 460, 231, 464, 233, 468, 235, 472, 237, 476, 239, 480, 241, 484, 243, 488, 245, 492, 247, 496, 249, 500, 251, 504, 253, 508, 255, 512, 257, 516, 259, 520, 261, 524, 263, 528, 265, 532, 267, 536, 269, 540, 271, 544, 273, 548, 275, 552, 277, 556, 279, 560, 281, 564, 283, 568, 285, 572, 287, 576, 289, 580, 291, 584, 293, 588, 295, 592, 297, 596, 299, 600, 301, 604, 303, 608, 305, 612, 307, 616, 309, 620, 311, 624, 313, 628, 315, 632, 317, 636, 319, 640, 321, 644, 323, 648, 325, 652, 327, 656, 329, 660, 331, 664, 333, 668, 335, 672, 337, 676, 339, 680, 341, 684, 343, 688, 345, 692, 347, 696, 349, 700, 351, 704, 353, 708, 355, 712, 357, 716, 359, 720, 361, 724, 363, 728, 365, 732, 367, 736, 369, 740, 371, 744, 373, 748, 375, 752, 377, 756, 379, 760, 381, 764, 383, 768, 385, 772, 387, 776, 389, 780, 391, 784, 393, 788, 395, 792, 397, 796, 399, 800, 401, 804, 403, 808, 405, 812, 407, 816, 409, 820, 411, 824, 413, 828, 415, 832, 417, 836, 419, 840, 421, 844, 423, 848, 425, 852, 427, 856, 429, 860, 431, 864, 433, 868, 435, 872, 437, 876, 439, 880, 441, 884, 443, 888, 445, 892, 447, 896, 449, 900, 451, 904, 453, 908, 455, 912, 457, 916, 459, 920, 461, 924, 463, 928, 465, 932, 467, 936, 469, 940, 471, 944, 473, 948, 475, 952, 477, 956, 479, 960, 481, 964, 483, 968, 485, 972, 487, 976, 489, 980, 491, 984, 493, 988, 495, 992, 497, 996, 499, 1000, 501, 1004, 503, 1008, 505, 1012, 507, 1016, 509, 1020, 511, 1024

Recall my “rational number” hint, and add to it an additional one – “irreducible fraction”. With them, and pairs of numbers, you should know what to do next. From there, with a key bit of preceding trickery, a common function fitting technique like Newton’s forward difference method, can find the final piece of the puzzle. (Additional clue: you don’t need nearly as many terms as I’ve included in these posts to solve the puzzle).

Of course, this all seems obvious to me, because I already know how I generated the original sequences. I’ll leave it to legitimate puzzle solvers who don’t already know its answer.

[Turtle]

Originally Posted by CraigD

As I hinted, modest surmised a couple of days ago, and I believe Turtle’s suggesting, a step in how I generated the sequences in post #1 was to sum the digits of the members of another integer sequence. This step of the puzzle solved, we can rewrite the sequences (for consistency’s sake, their first 512 terms):

[ lists removed for brevity's sake ]

Recall my “rational number” hint, and add to it an additional one – “irreducible fraction”. With them, and pairs of numbers, you should know what to do next. From there, with a key bit of preceding trickery, a common function fitting technique like Newton’s forward difference method, can find the final piece of the puzzle. (Additional clue: you don’t need nearly as many terms as I’ve included in these posts to solve the puzzle).

Of course, this all seems obvious to me, because I already know how I generated the original sequences. I’ll leave it to legitimate puzzle solvers who don’t already know its answer.

that is a lot like the joke about the ceiling to me; over my head. i'll keep trying to find an intuitive opening though. i have the world's only i-stick and i know how to use it.

on an aside from earlier, your new link on irreducible fraction stubs to "vulgar fraction" and there i find a strong case for my earlier mistake/misunderstanding of your equation in Latex. what is standard in one context may not be so in another. adding multiplier symbols to your equation would not be incorrect and would remove all doubt as to what is intended.

Originally Posted by Wiki

Mixed numbers
A mixed number is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+"; for example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: ...

[CraigD]

The “generalization of this post” I mentioned in post #1 sprang from noticing that, in the two box paradox thread, the original post sets up the paradox with a description stating “one box has 3 times more than the other”, while the the wikipedia article “two envelopes problem” sets it up with one stating “one envelope contains twice [(2 times)] as much as the other”.

This leads, in the “find the expected value” step of the paradox setup, to different results: for 2 times, for 3.

Consider what the expected value is for any “contains N times as much” integer. You’ll get this rational number sequence:


My puzzle sequence is simply the bit counts of the numerators of that sequence, the second “clue” sequence the denominatiors.

I though these were some pretty sequence, and a tricky puzzle for a couple of reasons:

The sequences increase on average, but not monotonically. The sequences of the bitcounts has periodic 2 or 1 terms, while the numerator and denominator sequences alternate between high and low values.

Given the sequence , the “workhorse” methods of finding a polynomial that fits it, , forward differences (specifically, I usually use Lagrange’s method, though there are others), doesn’t work. The differences never become zero. If you change it by multiplying each term by its position in the sequence to get this one

the difference method quickly finds the polynomial .

This last trick – multiplying each term of a sequence by its position (or ) seems a pretty valuable one in “find the polynomial that generated this sequence” type problems, as, if you figure out how to select the right it allows the difference method to succeed for polynomials with negative and/or non-integer exponents, like this puzzle's

, or, I think, more spectacular ones like

.