分球问题

王守恩

编号为1, 2, 3, ..., n的n个球,分成A,B两堆。满足：
①,A堆要有最大编号,  ②,B堆数目要大于A堆。譬如：
a(1)=0, 无法分。
a(2)=0, 无法分。
a(3)=1, {3}+{1,2},
a(4)=1, {4}+{1,2,3},
a(5)=5, {5}+{1,2,3,4},{5,1}+{2,3,4},{5,2}+{1,3,4},{5,3}+{1,2,4},{5,4}+{1,2,3},
a(6)=6, {6}+{1,2,3,4,5},{6,1}+{2,3,4,5},{6,2}+{1,3,4,5},{6,3}+{1,2,4,5},{6,4}+{1,2,3,5},{6,5}+{1,2,3,4},
a(7)=22, {7},{7,1},{7,2},{7,3},{7,4},{7,5},{7,6},{7,1,2},{7,1,3},{7,1,4},{7,1,5},{7,1,6},{7,2,3},{7,2,4},{7,2,5},{7,2,6},{7,3,4},{7,3,5},{7,3,6},{7,4,5},{7,4,6},{7,5,6}——把B堆省略了。
a(8)=29, {8},{8,1},{8,2},{8,3},{8,4},{8,5},{8,6},{8,7},{8,1,2},{8,1,3},{8,1,4},{8,1,5},{8,1,6},{8,1,7},{8,2,3},{8,2,4},{8,2,5},{8,2,6},{8,2,7},{8,3,4},{8,3,5},{8,3,6},{8,3,7},{8,4,5},{8,4,6},{8,4,7},{8,5,6},{8,5,7},{8,6,7},

0, 0, 1, 1, 5, 6, 22, 29, ......——这是怎样的一串数？

northwolves

1. Table[Sum[Binomial[n - 1, k], {k, 0, (n - 3)/2}], {n, 50}]

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{0,0,1,1,5,6,22,29,93,130,386,562,1586,2380,6476,9949,26333,41226,106762,169766,431910,695860,1744436,2842226,7036530,11576916,28354132,47050564,114159428,190876696,459312152,773201629,1846943453,3128164186,7423131482,12642301534,29822170718,51046844836,119766321572,205954642534,480832549478,830382690556,1929894318332,3345997029244,7744043540348,13475470680616,31067656725032,54244942336114,124613686513778,218269673491780}

northwolves

$a_n=2^{n-2} \left(1-\frac{\Gamma \left(\frac{n}{2}\right)}{\sqrt{\pi } \Gamma \left(\frac{n+1}{2}\right)}\right)$

王守恩

northwolves 发表于 2025-7-3 21:21
{0,0,1,1,5,6,22,29,93,130,386,562,1586,2380,6476,9949,26333,41226,106762,169766,431910,695860,1744 ...

看懂了——A294175——有这串数——没有我们的这个条文——我们的这个条文还是简单些。

王守恩

同理——A226881——有这串数——没有我们的这个条文——我们的这个条文还是简单些。
编号为1, 2, 3, ..., n的n个球,分成A,B两堆。满足：
①,A堆要有最大编号,  ②,B堆数目要小于A堆。譬如：
a(1)=0, 无法分。
a(2)=0, 无法分。
a(3)=2, {3,1}+{2},{3,2}+{1},
a(4)=3, {4,1,2}+{1},{3}+{2},{3}+{3},
a(5)=10, {1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}——把A堆省略了。
a(6)=15, {1},{2},{3},{4},{5},{1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5},
a(7)=41,
a(8)=63,

得到A226881——0, 0, 2, 3, 10, 15, 41, 63, 162, 255, 637, 1023, 2509, 4095, 9907, 16383, 39202, 65535, 155381, 262143, 616665, 1048575, ...

northwolves

王守恩 发表于 2025-7-4 10:56
同理——A226881——有这串数——没有我们的这个条文——我们的这个条文还是简单些。
编号为1, 2, 3, ...,  ...

1. Table[Sum[Binomial[n - 1, k], {k, 1, (n - 1)/2}], {n, 50}]

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王守恩

northwolves 发表于 2025-7-4 12:28

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,

{  }参考的是这串数——A062178——a(n+1) = 2a(n)-a([n/2]) starting with a(0)=0 and a(1)=1.

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, 5314400, 10628132, 21254942, 42508562,

A062178——a(n+1) = 2a(n)-a([n/2]) starting with a(0)=0 and a(1)=1——这公式应该怎样编排(我不会)？ 谢谢！！！

王守恩

northwolves 发表于 2025-7-4 12:28

{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, 5314400, 10628132, 21254942, 42508562, 85014494, 170026358, 340047481}

1. a[n_ /; n < 2] = n; a[n_?OddQ] := 2 a[n - 1] - a[(n - 1)/2]; a[n_?EvenQ] := 2 a[n - 1] - a[(n - 2)/2]; Table[a[n], {n, 0, 34}]

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这个可以用吗？谢谢！

northwolves

王守恩 发表于 2025-7-4 19:04
{0, 1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 16 ...

1. a[n_]:=Sign@n*(2a[n-1]-a[n/2-1]);Array[a@#&,34]

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northwolves

或者

1. a[n_] := If[n == 0, 0, 2 a[n - 1] - a[n/2 - 1]]; Array[a@# &, 34]

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