数字串的通项公式

王守恩 · 发表于 2024-9-1 20:17:49

漂亮的数字串! OEIS——没有这些漂亮的数字串。 OEIS——不应该有这些漂亮的数字串吗？

数字串(1)。1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 111, 333, 555, 777, 999, 1111, 3333, 5555, 7777, 9999, 11111, 33333, 55555, 77777, 99999, 111111, 333333, 555555, 777777, 999999, 1111111, 3333333,
5555555, 7777777, 9999999, 11111111, 33333333, 55555555, 77777777, 99999999, 111111111, 333333333, 555555555, 777777777, 999999999, 1111111111, 3333333333, 5555555555, 7777777777, ...

数字串(2)。1, 4, 7, 11, 44, 77, 111, 444, 777, 1111, 4444, 7777, 11111, 44444, 77777, 111111, 444444, 777777, 1111111, 4444444, 7777777, 11111111, 44444444, 77777777, 111111111, 444444444,
777777777, 1111111111, 4444444444, 7777777777, 11111111111, 44444444444, 77777777777, 111111111111, 444444444444, 777777777777, 1111111111111, 4444444444444, 7777777777777, ...

数字串(3)。2, 5, 8, 22, 55, 88, 222, 555, 888, 2222, 5555, 8888, 22222, 55555, 88888, 222222, 555555, 888888, 2222222, 5555555, 8888888, 22222222, 55555555, 88888888, 222222222, 555555555,
888888888, 2222222222, 5555555555, 8888888888, 22222222222, 55555555555, 88888888888, 222222222222, 555555555555, 888888888888, 2222222222222, 5555555555555, 8888888888888, ...

数字串(4)。3, 6, 9, 33, 66, 99, 333, 666, 999, 3333, 6666, 9999, 33333, 66666, 99999, 333333, 666666, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 333333333, 666666666,
999999999, 3333333333, 6666666666, 9999999999, 33333333333, 66666666666, 99999999999, 333333333333, 666666666666, 999999999999, 3333333333333, 6666666666666, 9999999999999, ...

数字串(5)。1, 6, 11, 66, 111, 666, 1111, 6666, 11111, 66666, 111111, 666666, 1111111, 6666666, 11111111, 66666666, 111111111, 666666666, 1111111111, 6666666666, 11111111111,
66666666666, 111111111111, 666666666666, 1111111111111, 6666666666666, 11111111111111, 66666666666666, 111111111111111, 666666666666666, 1111111111111111, 6666666666666666, ...

数字串(6)。2, 7, 22, 77, 222, 777, 2222, 7777, 22222, 77777, 222222, 777777, 2222222, 7777777, 22222222, 77777777, 222222222, 777777777, 2222222222, 7777777777, 22222222222,
77777777777, 222222222222, 777777777777, 2222222222222, 7777777777777, 22222222222222, 77777777777777, 222222222222222, 777777777777777, 2222222222222222, 7777777777777777, ...

数字串(7)。3, 8, 33, 88, 333, 888, 3333, 8888, 33333, 88888, 333333, 888888, 3333333, 8888888, 33333333, 88888888, 333333333, 888888888, 3333333333, 8888888888, 33333333333,
88888888888, 333333333333, 888888888888, 3333333333333, 8888888888888, 33333333333333, 88888888888888, 333333333333333, 888888888888888, 3333333333333333, 8888888888888888, ...

数字串(8)。4, 9, 44, 99, 444, 999, 4444, 9999, 44444, 99999, 444444, 999999, 4444444, 9999999, 44444444, 99999999, 444444444, 999999999, 4444444444, 9999999999, 44444444444,
99999999999, 444444444444, 999999999999, 4444444444444, 9999999999999, 44444444444444, 99999999999999, 444444444444444, 999999999999999, 4444444444444444, 9999999999999999, ...

数字串(9)。1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444,
55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 1111111, 2222222, 3333333, 4444444, 5555555, 6666666, 7777777, 8888888, 9999999, ...

王守恩 · 发表于 2024-9-4 08:55:56

王守恩发表于 2024-9-1 20:17
漂亮的数字串! OEIS——没有这些漂亮的数字串。 OEIS——不应该有这些漂亮的数字串吗？

数字串(1)。1, 3 ...

511#的通项公式出不来。

0, 1, 1, 2, 3, 5, 8, 10, 11, ...,通项公式就更难了。

0, 1, 1, 2, 3, 5, 8,

10, 11, 11, 22, 33, 55, 88,

100, 111, 111, 222, 333, 555, 888,

1000, 1111, 1111, 2222, 3333, 5555, 8888,

10000, 11111, 11111, 22222, 33333, 55555, 88888,

100000, 111111, 111111, 222222, 333333, 555555, 888888,

1000000, 1111111, 1111111, 2222222, 3333333, 5555555, ...

northwolves · 发表于 2024-9-5 08:42:38

数字串(1)。1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 111, 333, 555, 777, 999, 1111, 3333, 5555, 7777, 9999, 11111, 33333, 55555, 77777, 99999, 111111, 333333, 555555, 777777, 999999, 1111111, 3333333,
5555555, 7777777, 9999999, 11111111, 33333333, 55555555, 77777777, 99999999, 111111111, 333333333, 555555555, 777777777, 999999999, 1111111111, 3333333333, 5555555555, 7777777777, ...

$a_n=\frac{1+2((n-1)%5)}{9} *(10^{\lfloor \frac{n+4}{5}\rfloor }-1)$

数字串(9)。1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444,
55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 1111111, 2222222, 3333333, 4444444, 5555555, 6666666, 7777777, 8888888, 9999999, ...

$a_n=\left(\frac{n}{9}-\lfloor \frac{n-1}{9}\rfloor \right)*(10^{\lfloor \frac{n+8}{9}\rfloor }-1)$

其余同理。

王守恩 · 发表于 2024-9-5 11:05:12

漂亮的数字串! OEIS——没有这些漂亮的数字串。

数字串(5)。1, 6, 11, 66, 111, 666, 1111, 6666, 11111, 66666, 111111, 666666, 1111111, 6666666, 11111111, 66666666, 111111111, 666666666, 1111111111, 6666666666, 11111111111,

Table[Floor[(1 (1 - Cos[n Pi])*10^((n + 1)/2) + 6 (1 + Cos[n Pi]) 10^(n/2))/18], {n, 20}]

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数字串(6)。2, 7, 22, 77, 222, 777, 2222, 7777, 22222, 77777, 222222, 777777, 2222222, 7777777, 22222222, 77777777, 222222222, 777777777, 2222222222, 7777777777, 22222222222,

Table[Floor[(2 (1 - Cos[n Pi])*10^((n + 1)/2) + 7 (1 + Cos[n Pi]) 10^(n/2))/18], {n, 20}]

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数字串(7)。3, 8, 33, 88, 333, 888, 3333, 8888, 33333, 88888, 333333, 888888, 3333333, 8888888, 33333333, 88888888, 333333333, 888888888, 3333333333, 8888888888, 33333333333,

Table[Floor[(3 (1 - Cos[n Pi])*10^((n + 1)/2) + 8 (1 + Cos[n Pi]) 10^(n/2))/18], {n, 20}]

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数字串(8)。4, 9, 44, 99, 444, 999, 4444, 9999, 44444, 99999, 444444, 999999, 4444444, 9999999, 44444444, 99999999, 444444444, 999999999, 4444444444, 9999999999, 44444444444,

Table[Floor[(4 (1 - Cos[n Pi])*10^((n + 1)/2) + 9 (1 + Cos[n Pi]) 10^(n/2))/18], {n, 20}]

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5, 8, 55, 88, 555, 888, 5555, 8888, 55555, 88888, 555555, 888888, 5555555, 8888888, 55555555, 88888888, 555555555, 888888888, 5555555555, 8888888888, 55555555555, 88888888888, 555555555555, 888888888888,

Table[Floor[(5 (1 - Cos[n Pi])*10^((n + 1)/2) + 8 (1 + Cos[n Pi]) 10^(n/2))/18], {n, 24}]

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13, 35, 133, 355, 1333, 3555, 13333, 35555, 133333, 355555, 1333333, 3555555, 13333333, 35555555, 133333333, 355555555, 1333333333, 3555555555, 13333333333, 35555555555, 133333333333, 355555555555, 1333333333333,

Table[Floor[(12(1 -Cos[n Pi])*10^((n + 1)/2) +32(1 + Cos[n Pi]) 10^(n/2))/18], {n, 23}]

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王守恩 · 发表于 2024-9-5 14:28:58

数字串(5)。1, 6, 11, 66, 111, 666, 1111, 6666, 11111, 66666, 111111, 666666, 1111111, 6666666, 11111111, 66666666, 111111111, 666666666, 1111111111, 6666666666, 11111111111,

Table[((5 b - 4) (10^a - 1))/9, {a, 16}, {b, 2}] // Flatten

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数字串(6)。2, 7, 22, 77, 222, 777, 2222, 7777, 22222, 77777, 222222, 777777, 2222222, 7777777, 22222222, 77777777, 222222222, 777777777, 2222222222, 7777777777, 22222222222,

Table[((5 b - 3) (10^a - 1))/9, {a, 16}, {b, 2}] // Flatten

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数字串(7)。3, 8, 33, 88, 333, 888, 3333, 8888, 33333, 88888, 333333, 888888, 3333333, 8888888, 33333333, 88888888, 333333333, 888888888, 3333333333, 8888888888, 33333333333,

Table[((5 b - 2) (10^a - 1))/9, {a, 16}, {b, 2}] // Flatten

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数字串(8)。4, 9, 44, 99, 444, 999, 4444, 9999, 44444, 99999, 444444, 999999, 4444444, 9999999, 44444444, 99999999, 444444444, 999999999, 4444444444, 9999999999, 44444444444,

Table[((5 b - 1) (10^a - 1))/9, {a, 16}, {b, 2}] // Flatten

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王守恩 · 发表于 2024-9-6 10:19:09

northwolves 发表于 2024-9-5 08:42
数字串(1)。1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 111, 333, 555, 777, 999, 1111, 3333, 5555, 7777, 9999, ...

再来一串？谢谢！
{1, 3, 14, 37, 148, 370, 1481, 3703, 14814, 37037, 148148, 370370, 1481481, 3703703, 14814814, 37037037, 148148148, 370370370, 1481481481, 3703703703, 14814814814, 37037037037, 148148148148,
370370370370, 1481481481481, 3703703703703, 14814814814814, 37037037037037, 148148148148148, 370370370370370, 1481481481481481, 3703703703703703, 14814814814814814, 37037037037037037,

补充内容 (2024-9-8 16:29):
a(n)=Floor[((5 + Sqrt[40] + (5 - Sqrt[40]) Cos[n Pi]) (10^(n/2)) )/27]

王守恩 · 发表于 2024-9-7 02:53:11

简单=漂亮！漂亮=简单！
2, 8, 28, 85, 285, 857, 2857, 8571, 28571, 85714, 285714, 857142, 2857142, 8571428, 28571428, 85714285, 285714285, 857142857, 2857142857, 8571428571, 28571428571, 85714285714, ......

补充内容 (2024-9-8 16:26):
a(n)=Floor[((Sqrt[10] + 3 - Cos[n Pi] (Sqrt[10] - 3)) (10^(n/2)) )/7]

王守恩 · 发表于 2024-9-8 09:38:34

已知三角形两条高分别为 ha(整数) 和 hb(整数), 求第三条高 hc(整数) 的取值范围。

ha=1,hb=2,hc有1个解。hc=1,{{a -> 8/Sqrt[15], b -> 4/Sqrt[15],  c -> 8/Sqrt[15]}},
ha=2,hb=3,hc有4个解。hc=2,{{a -> 9/(2 Sqrt[2]), b -> 3/Sqrt[2], c -> 9/(2 Sqrt[2])}},
                                 hc=3,{{a -> 12/Sqrt[7], b -> 8/Sqrt[7], c -> 8/Sqrt[7]}},
                                 hc=4,{{a -> 144/Sqrt[455], b -> 96/Sqrt[455], c -> 72/Sqrt[455]}},
                                 hc=5,{{a -> 900/Sqrt[6479], b -> 600/Sqrt[6479], c -> 360/Sqrt[6479]}},
ha=3,hb=4,hc有10个解。hc=2,{{a -> 4.50055, b -> 3.37541, c -> 6.75082}},
                                 hc=3,{{a -> 4.31488, b -> 3.23616, c -> 4.31488}},
                                 hc=4,{{a -> 5.36656, b -> 4.02492, c -> 4.02492}},
                                 hc=5,{{a -> 6.69152, b -> 5.01864, c -> 4.01491}},
                                 hc=6,{{a -> 8.26236, b -> 6.19677, c -> 4.13118}},
                                 hc=7,{{a -> 10.1587, b -> 7.61905, c -> 4.35374}},
                                 hc=8,{{a -> 12.5581, b -> 9.41861, c -> 4.7093}},
                                 hc=9,{{a -> 15.8406, b -> 11.8804, c -> 5.28018}},
                                 hc=10,{{a -> 20.9857, b -> 15.7393, c -> 6.29572}},
                                 hc=11,{{a -> 31.8450, b -> 23.8837, c -> 8.685}},
ha=4,hb=5,hc有17个解。
ha=5,hb=6,hc有27个解。
ha=6,hb=7,hc有38个解。
ha=7,hb=8,hc有52个解。
ha=8,hb=9,hc有67个解。
.....
{1, 4, 10, 17, 27, 38, 52, 67, 85, 104, 126, 149, 175, 202, 232, 263, 297, 332, 370, 409, 451, 494, 540, 587, 637, 688, 742, 797, 855, 914, 976, 1039, 1105, 1172, 1242, 1313, 1387, 1462, 1540,
1619, 1701, 1784, 1870, 1957, 2047, 2138, 2232, 2327, 2425, 2524, 2626, 2729, 2835, 2942, 3052, 3163, 3277, 3392, 3510, 3629, 3751, 3874, 4000, 4127, 4257, 4388, 4522, 4657, 4795, 4934,
5076, 5219, 5365, 5512, 5662, 5813, 5967, 6122, 6280, 6439, 6601, 6764, 6930, 7097, 7267, 7438, 7612, 7787, 7965, 8144, 8326, 8509, 8695, 8882, 9072, 9263, 9457, 9652, 9850, 10049, ......}

Table[Floor[(2 n^2 + n - 1)/2], {n, 99}]

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OEIS有这串数——A213398——可没有这样的索引。

王守恩 · 发表于 2024-9-8 18:04:13

a, b, c 是3个不同的正整数, 满足 n + a*b + b*c + c*a = a*b*c, n = 1, 2, 3, 4, 5, 6, ...
n = 1, a+b+c=7+3+2=12(最小),
n = 2, a+b+c=5+4+2=11(最小),
n = 3, a+b+c=9+3+2=14(最小),
n = 4, a+b+c=6+4+2=12(最小),
n = 5, a+b+c=11+3+2=16(最小),
n = 6, a+b+c=7+4+2=13(最小),
n = 7, a+b+c=13+3+2=18(最小),
n = 8, a+b+c=6+5+2=13(最小),
n = 9, a+b+c=15+3+2=20(最小),
n = 10, a+b+c=9+4+2=15(最小),
n = 11, a+b+c=7+5+2=14(最小),
n = 12, a+b+c=10+4+2=16(最小),
n = 13, a+b+c=5+4+3=12(最小),
n = 14, a+b+c=8+5+2=15(最小),
n = 15, a+b+c=21+3+2=26(最小),
n = 16, a+b+c=7+6+2=15(最小),
n = 17, a+b+c=9+5+2=16(最小),
n = 18, a+b+c=6+4+3=13(最小),
n = 19, a+b+c=25+3+2=30(最小),
n = 20, a+b+c=8+6+2=16(最小),
n = 21, a+b+c=27+3+2=32(最小),
n = 22, a+b+c=15+4+2=21(最小),
n = 23, a+b+c=7+4+3=14(最小),
n = 24, a+b+c=9+6+2=17(最小),
n = 25, a+b+c=31+3+2=36(最小),

12, 11, 14, 12, 16, 13, 18, 13, 20, 15, 14, 16, 12, 15, 26, 15, 16, 13, 30, 16, 32, 21, 14, 17, 36, 17, 14, 15, 20, 25, 18, 19, 16, 15, 22, 19, 48, 17, 50, 21, 16, 31, 18, 20, 16, 15, 26, 17, 60, 21, 22,
21, 20, 17, 18, 22, 16, 21, 22, 26, 24, 19, 18, 27, 32, 23, 18, 17, 20, 45, 26, 19, 24, 17, 86, 21, 36, 19, 18, 25, 20, 51, 22, 25, 96, 25, 26, 18, 20, 19, 30, 26, 20, 27, 26, 31, 24, 29, 22, 21, 20, 19,
18, 25, 116, 21, 46, 19, 120, 31, 22, 21, 28, 67, 30, 20, 24, 27, 22, 41, 36, 19, 22, 30, 20, 25, 138, 29, 32, 21, 30, 23, 24, 23, 26, 30, 56, 23, 30, 31, 20, 21, 34, 22, 24, 31, 158, 21, 20, 31, 42, ...

Table[Min[a + b + c /. Solve[{n + a b + b c + c a == a b c, a > b > c > 0}, {a, b, c}, Integers]], {n, 151}]

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特别地。
a, b, c 是3个不同的正整数, 满足 n + a*b + b*c + c*a = a*b*c, n = 1, 2, 3, 4, 5, 6, ...
当 n = n, a + b + c = (n+6) + 3 + 2 = n + 11 时, n + a*b + b*c + c*a = n + (n+6)*3 + 3*2 + 2*(n+6) = (n+6)*3*2 = a*b*c (最大值)。

王守恩 · 发表于 2024-9-9 05:53:02

A007018.
n个单位分数(可以相同,可以不同)的和S(n), 满足2条 : 1, < 1; 2, 尽量大。
S (1) = 1/2 = 1/2.
S (2) = 1/3 + 1/2 = 5/6.
S (3) = 1/7 + 1/3 + 1/2 = 41/42.
S (4) = 1/43 + 1/7 + 1/3 + 1/21/2 = 1805/1806.
S (5) = 1/1807 + 1/43 + 1/7 + 1/3 + 1/2 = 3263441/3263442.
S (6) = 1/3263443 + 1/1807 + 1/43 + 1/7 + 1/3 + 1/2 = 10659956950605/10650056950806.
S (7) = 1/10650056950807 + 1/3263443 + 1/1807 + 1/43 + 1/7 + 1/3 + 1/2 = 113423713055421844361000441/113423713055421844361000442.
......

{1, 2, 6, 42, 1806, 3263442, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806,
165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185442,
273924503086030314234102342916746862811943643675809146279473679416086920262269936343321184045824386349295487372839923697584879743063177305807538834294603449564100770347613304760167394546498283855415...}

NestList[#^2 + # &, 1, 10]

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[原创] 数字串的通项公式

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