分球问题

王守恩

northwolves 发表于 2025-7-4 22:41

9#,10#显示不了。我可以先丢一丢。抓住机会赶快说重点,难题在7#：1, 3, 9, 23, 59, 135, 317, 713, 1607, 3527, 7745, 16769, 36235, 77607, 165825, 352317, 746685, 1576173, 3319593, ...可以有吗？

a (1) = 1 = {1} + 1 (2 - 1 - 1) = 1,
a (2) = 1 + 2 = {1 + 2} + 2 (3 - 1 - 2) = 3,
a (3) = 2 + 3 + 4 = {1 + 2 + 3} + 3 (5 - 2 - 2) = 9,
a (4) = 4 + 5 + 6 + 8 = {1 + 2 + 3 + 5} + 4 (8 - 2 - 3) = 23,
a (5) = 9 + 10 + 11 + 13 + 16 = {1 + 2 + 3 + 5 + 8} + 5 (14 - 3 - 3) = 59,
a (6) = 18 + 19 + 20 + 22 + 25 + 31 = {1 + 2 + 3 + 5 + 8 + 14} + 6 (25 - 3 - 5) = 135,
a (7) = 38 + 39 + 40 + 42 + 45 + 51 + 62 = {1 + 2 + 3 + 5 + 8 + 14 + 25} + 7 (47 - 5 - 5) = 317,
a (8) = 77 + 78 + 79 + 81 + 84 + 90 + 101 + 123 = {1 + 2 + 3 + 5 + 8 + 14 + 25 + 47} + 8 (89 - 5 - 8) = 713,
a (9) = 158 + 159 + 160 + 162 + 165 + 171 + 182 + 204 + 246 = {1 + 2 + 3 + 5 + 8 + 14 + 25 + 47 + 89} + 9 (173 - 8 - 8) = 1607,

\(\D a(n)=\sum_{k=1}^n\ b_{k}+n\cdot\big(b_{n+1}-b_{\lfloor(n+1)/2\rfloor}{}-b_{\lfloor(n+2)/2\rfloor}\big)\)

其中b(n)=0, 1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936, 2657534, —— A062178——a(n+1) = 2a(n)-a([n/2]) starting with a(0)=0 and a(1)=1

northwolves

王守恩 发表于 2025-7-5 05:53
9#,10#显示不了。我可以先丢一丢。抓住机会赶快说重点,难题在7#：1, 3, 9, 23, 59, 135, 317, 713, 1607,  ...

这个公式太难看了

northwolves

1. b[n_] := If[n < 2, n, 2 b[n - 1] - b[Floor[(n - 1)/2]]];
   
2. a[n_] := Sum[b[k], {k, n}] + n*(b[n + 3] - b[n + 1] - b[n + 2]);
   
3. Array[b@# &, 34]

复制代码

{1,2,3,5,8,14,25,47,89,173,338,668,1322,2630,5235,10445,20843,41639,83189,166289,332405,664637,1328936,2657534,5314400,10628132,21254942,42508562,85014494,170026358,340047481,680089727,1360169009,2720327573}

王守恩

northwolves 发表于 2025-7-5 10:37
{1,2,3,5,8,14,25,47,89,173,338,668,1322,2630,5235,10445,20843,41639,83189,166289,332405,664637,132 ...

b(n)——1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843,——A062178——没有通项公式——你得为广大数学爱好者想想——添一个通项公式。

a(n)——1, 3, 9, 23, 59, 135, 317, 713, 1607, 3527, 7745, 16769, 36235, 77607, 165825, 352317, 746685, 1576173, 3319593, 6970845, 14608551, 30544467,——OEIS没有——你应该去申报。

补充内容 (2025-7-7 04:19):
b[n_] := If[n < 2, n, 2 b[n - 1] - b[Floor[(n - 1)/2]]]; Array[b@# &, 34]

补充内容 (2025-7-7 04:25):
a[n_] := Sum[b[k], {k, n}] + n*(b[n + 3] - b[n + 1] - b[n + 2]);
b[n_] := If[n < 2, n, 2 b[n - 1] - b[Floor[(n - 1)/2]]]; Array[
a@# &, 30]

northwolves

王守恩 发表于 2025-7-5 12:08
b(n)——1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843,——A062178— ...

a(n)有什么实际中的意义呢？王老师您自己提交吧

王守恩

编号为1,2,3,...,n的n个球,  重量(正整数)依次为f(1)<f(2)<f(3)<...<f(n)。

选若干数目的球并将所选的球分成两堆,无论哪堆有哪些球,一定满足以下条件:

①若两堆球的数目不同,则球较多的一堆一定比另一堆重。

②若两堆球的数目相同,则两堆中编号最大球所在的一堆一定比另一堆重。

求a(n)=f(1)+f(2)+f(3)+...+f(n)的最小值。譬如:
a(1)=1, 1={1}+0,
a(2)=3, 1+2={1,2}+0,
a(3)=9, 2+3+4={1,2,3}+1,
a(4)=23, 4+5+6+8={1,2,3,5}+3,
a(5)=59, 9+10+11+13+16={1,2,3,5,8}+8,
a(6)=135, 18+19+20+22+25+31={1,2,3,5,8,14}+17,
a(7)=317, 38+39+40+42+45+51+62={1,2,3,5,8,14,25}+37,
a(8)=713, 77+78+79+81+84+90+101+123={1,2,3,5,8,14,25,47}+76,
a(9)=1607, 158+159+160+162+165+171+182+204+246={1,2,3,5,8,14,25,47,89}+157,
a (1) = 1 = {1} + 1 (2 - 1 - 1) = 1,
a (2) = 1 + 2 = {1 + 2} + 2 (3 - 1 - 2) = 3,
a (3) = 2 + 3 + 4 = {1 + 2 + 3} + 3 (5 - 2 - 2) = 9,
a (4) = 4 + 5 + 6 + 8 = {1 + 2 + 3 + 5} + 4 (8 - 2 - 3) = 23,
a (5) = 9 + 10 + 11 + 13 + 16 = {1 + 2 + 3 + 5 + 8} + 5 (14 - 3 - 3) = 59,
a (6) = 18 + 19 + 20 + 22 + 25 + 31 = {1 + 2 + 3 + 5 + 8 + 14} + 6 (25 - 3 - 5) = 135,
a (7) = 38 + 39 + 40 + 42 + 45 + 51 + 62 = {1 + 2 + 3 + 5 + 8 + 14 + 25} + 7 (47 - 5 - 5) = 317,
a (8) = 77 + 78 + 79 + 81 + 84 + 90 + 101 + 123 = {1 + 2 + 3 + 5 + 8 + 14 + 25 + 47} + 8 (89 - 5 - 8) = 713,
a (9) = 158 + 159 + 160 + 162 + 165 + 171 + 182 + 204 + 246 = {1 + 2 + 3 + 5 + 8 + 14 + 25 + 47 + 89} + 9 (173 - 8 - 8) = 1607,
这题目看起来挺复杂的,实际上只要满足一个条件。
a (1) = 1,
a (2) = 1 - 2,
a (3) = 2 + 3 - 4 =1, 2个比1个,
a (4) = 4 - 5 - 6 + 8 = 1, 2个比2个,
a (5) = 9 + 10 + 11 - 13 - 16 =1, 3个比2个,
a (6) = 18 + 19 - 20 - 22 - 25 + 31 =1, 3个比3个,
a (7) = 38 + 39 + 40 + 42 - 45 - 51 - 62 =1, 4个比3个,
a (8) = 77 + 78 + 79 - 81 - 84 - 90 - 101 + 123 =1, 4个比4个,
a (9) = 158 + 159 + 160 + 162 + 165 - 171 - 182 - 204 - 246 =1, 5个比4个,

好玩！！！好像跟《求{1, 2, ..., 100}没有等和对的最大子集》有联系。好玩才是源动力！！！
2串数。1, 1, 2, 4, 9, 18, 38, 77, 158, ......。1, 1, 2, 3, 6, 11, 22, 42, .....。
搞复杂了——不能搞复杂——还是回归13#公式——只要有13#公式就可以了。

王守恩

northwolves 发表于 2025-7-5 14:21
a(n)有什么实际中的意义呢？王老师您自己提交吧

去不去OEIS求助——那是你的事——我是丢了——差不多搞懂了。

若去——这个必须有——最好的说明——一目了然。

a(1)=1, {1},
a(2)=3, {1,2},
a(3)=9, {2,3,4},
a(4)=23,{4,5,6,8},
a(5)=59, {9,10,11,13,16},
a(6)=135, {18,19,20,22,25,31},
a(7)=317, {38,39,40,42,45,51,62},
a(8)=713, {77,78,79,81,84,90,101,123},
a(9)=1607, {158,159,160,162,165,171,182,204,246},

王守恩

northwolves 发表于 2025-7-4 12:28

\(将1,2,3,...,10十个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{10},约定i-A≤a_{i}≤i+B。求满足条件的排列数量。\)

A,B是不同或相同的正整数——OEIS不一定有这些数字串——手工有点难——可有好的方法。谢谢！

王守恩

来点简单的。

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+1。求满足条件的排列数量。\)
{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]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+2。求满足条件的排列数量。\)
{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]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+3。求满足条件的排列数量。\)
{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]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+4。求满足条件的排列数量。\)
{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]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+5。求满足条件的排列数量。\)
{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]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+6。求满足条件的排列数量。\)
{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]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+7。求满足条件的排列数量。\)
{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]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-1≤a_{i}≤i+8。求满足条件的排列数量。\)
{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]

\(备忘:i-A≤a_{i}≤i+B\ 与\ i-B≤a_{i}≤i+A\ 答案是一样的。A,B是不同或相同的正整数。\)

王守恩

这题目有点难。目的很明确——把18#的通解公式引出来！！！

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-2≤a_{i}≤i+2。求满足条件的排列数量。\)
1, 2, 6, 14, 31, 73, 172, 400, 932, 2177, 5081, 11854, 27662, 64554, 150639, 351521, 820296, 1914208, 4466904, 10423761, 24324417, 56762346, 132458006, 309097942, 721296815, 1683185225, 3927803988},
LinearRecurrence[{2, 0, 2, 0, -1}, {1, 0, 0, 0, 1}, 33]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-2≤a_{i}≤i+3。求满足条件的排列数量。\)
1, 2, 6, 18, 46, 115, 301, 792, 2068, 5380, 14020, 36581, 95413, 248786, 648714, 1691686, 4411530, 11503991, 29998953, 78228640, 203998184, 531969064, 1387222648, 3617479225, 9433351129, 24599481138},
LinearRecurrence[{1, 2, 3, 5, 6, -1, -1, 0, -1, -1}, {1, 0, 0, 0, 0, 1, 1, 2, 6, 18}, 34]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-2≤a_{i}≤i+4。求满足条件的排列数量。\)
1, 2, 6, 18, 54, 146, 391, 1081, 3004, 8320, 22984, 63424, 175176, 484113, 1337721, 3695886, 10210702, 28209954, 77940078, 215337554, 594943087, 1643728129, 4541349672, 12547013504, 34665373744, 95774808224},
LinearRecurrence[{1, 2, 4, 6, 10, 12, -4, -6, -6, 0, -2, -2, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 1, 1, 2, 6, 18, 54, 146, 391, 1081}, 40]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-2≤a_{i}≤i+5。求满足条件的排列数量。\)
1, 2, 6, 18, 54, 162, 454, 1267, 3613, 10344, 29572, 84436, 240868, 686884, 1959636, 5592181, 15957717, 45533682, 129922090, 370708166, 1057755082, 3018154342, 8611878218, 24572725639, 70114579881, 200061418144},
LinearRecurrence[{1, 2, 4, 7, 13, 22, 28, -4, -6, -6, 0, -4, -10, -10, 0, 1, 1, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 18, 54, 162, 454, 1267, 3613, 10344, 29572, 84436, 240868}, 54]

\(将1,2,3,...,n这n个数重新排列,得到新序列a_{1},a_{2},a_{3},...,a_{n},约定i-2≤a_{i}≤i+6。求满足条件的排列数量。\)
1, 2, 6, 18, 54, 162, 486, 1394, 3991, 11593, 33772, 98320, 286072, 831952, 2418664, 7030816, 20441944, 59441521, 172843609, 502580846, 1461344622, 4249102850, 12354982862, 35924300898, 104456501102, 303726483778},
LinearRecurrence[{1, 2, 4, 8, 14, 26, 44, 56, -11, -19, -28, -28, 0, -8, -20, -20, 0, 5, 11, 10, 0, 0, 2, 2, 0, 0, -1, -1}, {48, -40, 32, -24, 17, -7, 6, -2, 2, -2, 2, -2, 2, -1, 1, 0, 0, 0, 0, 0, 0, 0,1, 1, 2, 6, 18, 54}, 67]

\(备忘:i-A≤a_{i}≤i+B\ 与\ i-B≤a_{i}≤i+A\ 答案是一样的。A,B是不同或相同的正整数。\)