数字串的通项公式

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northwolves 发表于 2024-7-15 12:30
{{1,7},{2,11},{3,13},{4,14},{5,19},{6,21},{7,22},{8,25},{9,26},{10,28},{11,35},{12,37},{13,38},{ ...

这些按钮还是学不了。A115881——应该有通项公式的。

0, 0, 1, 0, 4, 2, 9, 1, 16, 8, 25, 4, 36, 18, 49, 9, 64, 32, 81, 16, 100, 50, 121, 25, 144, 72, 169, 36, 196, 98, 225, 49, 256, 128, 289, 64, 324, 162, 361, 81, 400, 200, 441, 100, 484, 242, 529, 121, 576,

northwolves

王守恩 发表于 2024-7-20 13:54
这些按钮还是学不了。A115881——应该有通项公式的。

0, 0, 1, 0, 4, 2, 9, 1, 16, 8, 25, 4, 36, 18, 49 ...

$a_n=(\lfloor \frac{n+6}{4}\rfloor -\lfloor \frac{n+1}{4}\rfloor )*\lfloor\frac{n+1}{3+(-1)^n}-\frac{1}{2}\rfloor ^2$

王守恩

northwolves 发表于 2024-7-23 10:31
$a_n=(\lfloor \frac{n+6}{4}\rfloor -\lfloor \frac{n+1}{4}\rfloor )*\lfloor\frac{n+1}{3+(-1)^n}-\fr ...

非负整数 a,b,c,d,  a,b,c 成等差数列,  b,c,d 成等比数列,  其中: d=a+n,  a,b,c,d 最大是多少？
n=1,无解,
n=2,无解,
n=3,无解,
n=4,a,b,c,d={0,1,2,4},
n=5,无解,
n=6,无解,
n=7,a,b,c,d={2,4,6,9},
n=8,a,b,c,d={0,2,4,8},
n=9,无解,
n=10,a,b,c,d={6,9,12,16},
n=11,无解,
n=12,a,b,c,d={0,3,6,12},
n=13,a,b,c,d={12,16,20,25},
n=14,a,b,c,d={4,8,12,18},
n=15,无解,
n=16,a,b,c,d={20,25,30,36},
n=17,无解,
n=18,无解,
n=19,a,b,c,d={30,36,42,49},
n=20,a,b,c,d={12,18,24,32},
n=21,a,b,c,d={6,12,18,27},
n=22,a,b,c,d={42,49,56,64},
n=23,无解,
n=24,a,b,c,d={0,6,12,24},
n=25,a,b,c,d={56,64,72,81},
n=26,a,b,c,d={24,32,40,50},
n=27,无解,
n=28,a,b,c,d={72,81,90,100},
n=29,无解,
n=30,a,b,c,d={18,27,36,48},
n=31,a,b,c,d={90,100,110,121},
n=32,a,b,c,d={40,50,60,72},
n=33,无解,
n=34,a,b,c,d={110,121,132,144},
n=35,a,b,c,d={10,20,30,45},
n=36,a,b,c,d={0,9,18,36},
n=37,a,b,c,d={132,144,156,169},
n=38,a,b,c,d={60.72,84,98},
n=39,a,b,c,d={36,48,60,75},
n=40,a,b,c,d={156,169,182,196},
......
得到一串数。
{a, a, a, 0, a, a, 2, 0, a, 6, a, 0, 12, 4, a, 20, a, a, 30, 12, 6, 42, a, 0, 56, 24, a, 72, a, 18, 90, 40, a, 110, 10, 0, 132, 60, 36, 156, a, 12, 182, 84, 4, 210, a, 60, 240, 112, a, 272, a, 10, 306, 144, 90, 342, a, 36, 380, 180, 18, 420, 60, 126, 462, 220}

1. Table[Max[a /. Solve[{(2 c)/(a + c) == (a + n)/c, 0 <= a <= c,  Mod[a + c, 2] == 0}, {a, c}, Integers]], {n, 1, 68}]

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这通项公式只能这样了吧？

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{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, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60}

1. Table[Ceiling[(n + Pi)/Power[E^Pi, (n)^-1]], {n, 60}]

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会有对不上的吗？！

王守恩

A054925——这通项公式会有问题吗？！

0, 0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53, 60, 68, 77, 86, 95, 105, 116, 127, 138, 150, 163, 176, 189, 203, 218, 233, 248, 264, 281, 298, 315, 333, 352, 371, 390, 410, 431, 452, 473, 495, 518, 541, 564, 588, 613, 638, 663, 689, ...

1. Table[Round[(n - 1)/(Power[E^4, (n)^-1] - 1)], {n, 60}]

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王守恩

求通项公式。

{0, 1, 0, 1, 2, 1, 2, 1, 2, 4, 2, 4, 2, 4, 2, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 9, 6, 9, 6, 9, 6, 9, 6, 9, 6, 9, 12, 9, 12, 9, 12, 9, 12, 9, 12, 9, 12, 9, 12, 16, 12, 16, 12, 16, 12, 16, 12, 16, 12,
16, 12, 16, 12, 16, 20, 16, 20, 16, 20, 16, 20, 16, 20, 16, 20, 16, 20, 16, 20, 16, 20, 25, 20, 25, 20, 25, 20, 25, 20, 25, 20, 25, 20, 25, 20, 25, 20, 25, 20, 25, 30, 25, 30, 25,
30, 25, 30, 25, 30, 25, 30, 25, 30, 25, 30, 25, 30, 25, 30, 25, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 30, 36, 42, 36, 42, 36, 42,
36, 42, 36, 42, 36, 42, 36, 42, 36, 42, 36, 42, 36, 42, 36, 42, 36, 42, 36, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49, 42, 49,
42, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 49, 56, 64, 56, 64, 56, 64, 56, 64, 56, 64, 56, 64, 56, 64,......,

0有2个。1有4个。2有6个。4有8个。6有10个。9有12个。12有14个。16有16个。20有18个。25有20个。30有22个。36有24个。42有26个。49有28个。56有30个。64有32个。

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0, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 31,
32, 32, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 56, 56, 57, 57, 58, 58,
59, 59, 60, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 67, 67, 68, 68, 69, 69, 70, 70, 71, 71, 72, 72, 73, 73, 74, 74, 75, 75, 76, 76, 77, 77, 78, 79, 79, 80, 80, 81, 81, 82, 82, 83, 83, 84, 84, 85,
85, 86, 86, 87, 87, 88, 88, 89, 89, 90, 90, 91, 92, 92, 93, 93, 94, 94, 95, 95, 96, 96, 97, 97, 98, 98, 99, 99, 100, 100, 101, 101, 102, 102, 103, 103, 104, 104, 105, 106, 106, 107, 107, 108, 108, ...}

A259549——有个通项公式(Jun 08 2019)——看我们的  !  !  !

1. Table[Round[(n + Sqrt[n])/2], {n, 0, 200}]

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王守恩

A035608——这个通项也可以——Table[Floor[(n - 1)/(Power[E, (n)^-1] - 1)], {n, 180}]
               
0, 1, 5, 10, 18, 27, 39, 52, 68, 85, 105, 126, 150, 175, 203, 232, 264, 297, 333, 370, 410, 451, 495, 540, 588, 637, 689, 742, 798, 855, 915, 976, 1040, 1105, 1173, 1242, 1314, 1387, 1463, 1540, 1620, 1701,
1785, 1870, 1958, 2047, 2139, 2232, 2328, 2425, 2525, 2626, 2730, 2835, 2943, 3052, 3164, 3277, 3393, 3510, 3630, 3751, 3875, 4000, 4128, 4257, 4389, 4522, 4658, 4795, 4935, 5076, 5220, 5365, 5513,
5662, 5814, 5967, 6123, 6280, 6440, 6601, 6765, 6930, 7098, 7267, 7439, 7612, 7788, 7965, 8145, 8326, 8510, 8695, 8883, 9072, 9264, 9457, 9653, 9850, 10050, 10251, 10455, 10660, 10868, 11077, ....

A014848——这个通项也可以——Table[Round[n/(Power[E, (n)^-1] - 1)], {n, 180}]       

1, 3, 8, 14, 23, 33, 46, 60, 77, 95, 116, 138, 163, 189, 218, 248, 281, 315, 352, 390, 431, 473, 518, 564, 613, 663, 716, 770, 827, 885, 946, 1008, 1073, 1139, 1208, 1278, 1351, 1425, 1502, 1580, 1661,
1743, 1828, 1914, 2003, 2093, 2186, 2280, 2377, 2475, 2576, 2678, 2783, 2889, 2998, 3108, 3221, 3335, 3452, 3570, 3691, 3813, 3938, 4064, 4193, 4323, 4456, 4590, 4727, 4865, 5006, 5148, 5293, 5439,
5588, 5738, 5891, 6045, 6202, 6360, 6521, 6683, 6848, 7014, 7183, 7353, 7526, 7700, 7877, 8055, 8236, 8418, 8603, 8789, 8978, 9168, 9361, 9555, 9752, 9950, 10151, 10353, 10558, 10764, 10973,  .........

王守恩

a(1)=2——用1个1与1个2组成的2位置数有2个。
12,
21,
a(2)=6——用2个1与2个2组成的4位置数有6个。
1122,
1212,
1221,
2112,
2121,
2211,
a(3)=20——用3个1与3个2组成的6位置数有20个。
111222,
112122,
112212,
112221,
121122,
121212,
121221,
122112,
122121,
122211,
a(4)=70——用4个1与4个2组成的8位置数有70个。
11112222,
11121222,
11122122,
11122212,
11122221,
11211222,
11212122,
11212212,
11212221,
11221122,
11221212,
11221221,
11222112,
11222121,
11222211,
12111222,
12112122,
12112212,
12112221,
12121122,
12121212,
12121221,
12122112,
12122121,
12122211,
12211122,
12211212,
12211221,
12212112,
12212121,
12212211,
12221112,
12221121,
12221211,
12222111,
a(5)——用5个1与5个2组成的10位置数有？个。
a(6)——用6个1与6个2组成的12位置数有？个。
a(7)——用7个1与7个2组成的14位置数有？个。
a(8)——用8个1与8个2组成的16位置数有？个。
a(9)——用9个1与9个2组成的18位置数有？个。



补充内容 (2024-8-9 10:02):
a(1)=2——用1个1与1个2组成的2位数有2个。

王守恩

A134030——2024年3月11日
0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18, 20, 23, 26, 28, 32, 35, 38, 42, 46, 49, 54, 58, 62, 67, 71, 76, 81, 86, 92, 97, 103, 109,
115, 121, 127, 134, 140, 147, 154, 161, 168, 176, 183, 191, 199, 207, 215, 223, 232, 240, 249, 258, 267, 277, 286, 296, 306,
316, 326, 336, 346, 357, 368, 379, 390, 401, 412, 424, 436, 447, 459, 472, 484, 496, 509, 522, 535, 548, 561, 575, 588, 602,
616, 630, 644, 659, 673, 688, 703, 718, 733, 748, 764, 780, 796, 812, 828, 844, 860, 877, 894, 911, 928, 945, 963, 980, 998}

1. Table[Round[Cot[Pi/n]*n/4], {n, 3, 112}]

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