求自然数列 A＝{1,2,6,7,9,10,11,…}，B={0,3,4,5,8,12,…} 的通项公式

SHEEN

有两个自然数数列A、B， A＝{1、2、6、7、9、10、11...} ，B={0、3、4、5、8、12...} 。

A中元素：2的奇数倍、B中元素乘以2加1两种组成，
B中元素：2的偶数倍、A中元素乘以2加1两种组成，

求A、B的通项公式？

northwolves

$A=16n+{1,2,6,7,9,10,11,14}$
$B=16n+{0,3,4,5,8,12,13,15}$

王守恩

northwolves 发表于 2021-10-28 18:07
$A=16n+{1,2,6,7,9,10,11,14}$
$B=16n+{0,3,4,5,8,12,13,15}$

电脑可以出来数字串，但出不了通项公式。

A={1, 2, 6, 7, 9, 10, 11, 14, 17, 18, 22, 25, 26, 27, 30, 31, 33, 34, 38, 39, 41, 42, 43, 46, 47, 49, 50,
54, 57, 58, 59, 62, 65, 66, 70, 71, 73, 74,  75, 78, 81, 82, 86, 89, 90, 91,  94, 97, 98, 102, 103, 105,
106, 107, 110, 111, 113, 114, 118, 121, 122, 123, 126, 127, 129, 130, 134, 135, 137, 138, 139, 142,
145, 146, 150, 153, 154, 155, 158, 159, 161, 162, 166, 167, 169, 170, 171, 174, 175, 177, 178, 182,
185, 186, 187, 190, 191, 193, 194, 198, 199, 201, 202, 203, 206, 209, 210, 214, 217, 218, 219, 222,
225, 226, 230, 231, 233, 234, 235, 238, 239, 241, 242, 246, 249, 250, 251, 254, 257, 258, 262, 263,
265, 266, 267, 270, 273, 274, 278, 281, 282, 283, 286, 287, 289, 290, 294, 295, 297, 298, 299, 302,
303, 305, 306, 310, 313, 314, 315, 318, 321, 322, 326, 327, 329, 330, 331, 334, 337, 338, 342, 345,
346, 347, 350, 353, 354, 358, 359, 361, 362, 363, 366, 367, 369, 370, 374, 377, 378, 379, 382, 385,
386, 390, 391, 393, 394, 395, 398, 401, 402, 406, 409, 410, 411, 414, 415, 417, 418, 422, 423, 425,
426, 427, 430, 431, 433, 434, 438, 441, 442, 443, 446, 447, 449, 450, 454, 455, 457, 458, 459, 462,
465, 466, 470, 473, 474, 475, 478, 481, 482, 486, 487, 489, 490, 491, 494, 495, 497, 498,......

B={0, 3, 4, 5, 8, 12, 13, 15, 16, 19, 20, 21, 23, 24, 28, 29, 32, 35, 36, 37, 40, 44, 45, 48, 51, 52, 53,
55, 56,  60, 61,  63,  64, 67, 68 , 69, 72, 76, 77, 79,  80, 83, 84, 85, 87, 88, 92, 93, 95, 96, 99, 100,
101, 104, 108, 109, 112, 115, 116, 117, 119, 120, 124, 125, 128, 131, 132, 133, 136, 140, 141, 143,
144, 147, 148, 149, 151, 152, 156, 157, 160, 163, 164, 165, 168, 172, 173, 176, 179, 180, 181, 183,
184, 188, 189, 192, 195, 196, 197, 200, 204, 205, 207, 208, 211, 212, 213, 215, 216, 220, 221, 223,
224, 227, 228, 229, 232, 236, 237, 240, 243, 244, 245, 247, 248, 252, 253, 255, 256, 259, 260, 261,
264, 268, 269, 271, 272, 275, 276, 277, 279, 280, 284, 285, 288, 291, 292, 293, 296, 300, 301, 304,
307, 308, 309, 311, 312, 316, 317, 319, 320, 323, 324, 325, 328, 332, 333, 335, 336, 339, 340, 341,
343, 344, 348, 349, 351, 352, 355, 356, 357, 360, 364, 365, 368, 371, 372, 373, 375, 376, 380, 381,
383, 384, 387, 388, 389, 392, 396, 397, 399, 400, 403, 404, 405, 407, 408, 412, 413, 416, 419, 420,
421, 424, 428, 429, 432, 435, 436, 437, 439, 440, 444, 445, 448, 451, 452, 453, 456, 460, 461, 463,
464, 467, 468, 469, 471, 472, 476, 477, 479, 480, 483, 484, 485, 488, 492, 493, 496, 499, 500, ........

SHEEN

northwolves 发表于 2021-10-28 18:07
$A=16n+{1,2,6,7,9,10,11,14}$
$B=16n+{0,3,4,5,8,12,13,15}$

你考察的数范围小了，扩大你考察的范围，你会发现你的表达式不能完整表达

mathe

A中8k+1,8k+2,8k+6,加上$u2^{2h}+2^{2h-1}-1 (h\ge 2) $
B中8k+3,8k+4,8k+8,加上$u2^{2h+1}+2^{2h}-1 (h\ge 2)$

SHEEN

mathe 发表于 2021-10-29 12:51
A中8k+1,8k+2,8k+6,加上$u2^{2h}+2^{2h-1}-1 (h\ge 2) $
B中8k+3,8k+4,8k+8,加上$u2^{2h+1}+2^{2h}-1 (h\g ...

麻烦详细说下这个表达式是什么意思，比如U是什么？h由什么决定值

王守恩

王守恩 发表于 2021-10-28 18:39
电脑可以出来数字串，但出不了通项公式。

A={1, 2, 6, 7, 9, 10, 11, 14, 17, 18, 22, 25, 26, 27,  ...

\(\D A(n)=2^k(n-mod(k,2))-2^{k-2}\cos(k\pi)-1\)     n=1,2,3,4,...     k=2,3,4,...
A={1, 2, 6, 7, 9, 10, 11, 14, 17, 18, 22, 25, 26, 27, 30, 31, 33, 34, 38, 39, 41, 42, 43, 46, 47, 49, 50,

\(\D B(n)=2^k(n+mod(k,2))-2^{k-2}\cos(k\pi)-1\)     n=0,1,2,3,...     k=2,3,4,...
B={0, 3, 4, 5, 8, 12, 13, 15, 16, 19, 20, 21, 23, 24, 28, 29, 32, 35, 36, 37, 40, 44, 45, 48, 51, 52, 53,

\(\D A(n)+B(n)=2^k\bigg\lceil\frac{n}{2}\bigg\rceil+2^{k-2}\cos(n\pi)-1\)         n=0,1,2,3,...      k=2,3,4,...
A+B={0, 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,...