今天是11月23日(斐波那契纪念日)

王守恩

神奇得数。

\(\frac{1}{(2\cos(\pi/5))^0}+\frac{1}{(2\cos(\pi/5))^1}+\frac{1}{(2\cos(\pi/5))^2}+\frac{1}{(2\cos(\pi/5))^3}+......=\frac{\sqrt{5}+1}{\sqrt{5}-1}\)

淘气小孩，在海滩玩耍发现一块贝壳，奔跑相告，

感染我们的是那种快乐，让我们跟着小孩一起奔跑！

王守恩

2025中国科技大学强基计划数学试题 ——第7题——扩散。

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+1。求满足条件的排列数量。\)
{1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352}
LinearRecurrence[{1, 1}, {0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+2。求满足条件的排列数量。\)
{1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080, 98950096, 181997601}
LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+3。求满足条件的排列数量。\)
{1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533, 28074040, 54114452, 104308960, 201061985, 387559437}
LinearRecurrence[{1, 1, 1, 1}, {0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+4。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351, 678355061}
LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+5。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728, 14247536, 28261168, 56058368, 111196417, 220567305, 437513522, 867844316}
LinearRecurrence[{1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+6。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600, 971364608}
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+7。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, 16190208, 32316160, 64504063, 128752121, 256993248, 512966536, 1023897200}
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+8。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729}
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 40]