三重内积(增加难度)的最小值

王守恩

来个估算公式。有底气些。——Table[A*Ceiling[A!^(n/A)], {A, 2, 6}, {n, 35}]——谢谢mathe——详见《三重内积的最小值》——6#。——谢谢mathe。

公式。{4, 4, 6, 8, 12, 16, 24, 32, 46, 64, 92, 128, 182, 256, 364, 512, 726, 1024, 1450, 2048, 2898, 4096, 5794, 8192, 11586, 16384, 23172, 32768, 46342, 65536, 92682, 131072, 185364, 262144, 370728},A029744
A—2。{3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, —— A029744——虽然还没有这条文。

公式。{6, 12, 18, 33, 60, 108, 198, 357, 648, 1179, 2142, 3888, 7065, 12840, 23328, 42390, 77028, 139968, 254340, 462165, 839808, 1526034, 2772987, 5038848, 9156195, 16637910, 30233088, 54937167, 99827460, 181398528},
A—3。{6, 10, 18, 33, 60, 108, 198, 360, 648, 1188, 2145, 3888, 7083, 12844, 23328, 42498, 77064, 139968, 254988, 462384, 839808, 1526769, 2774304, 5038848, 9160614, 16645824, 30233088, ——29#——A=1,2,3。

公式。{12, 20, 44, 96, 216, 472, 1044, 2304, 5100, 11288, 24984, 55296, 122392, 270896, 599588, 1327104, 2937368, 6501456, 14390088, 31850496, 70496740, 156034928, 345362068, 764411904, 1691921668},
A—4。{10, 20, 44, 96, 214, 472, 1043, 2304, 5136, 11328, 24993, 55296, 122624, 271040,——28#——A=1,2,3,4。

公式。{15, 35, 90, 235, 600, 1565, 4075, 10610, 27640, 72000, 187575, 488660, 1273040, 3316485, 8640000, 22508680, 58638960, 152764520, 397977705, 1036800000, 2701041385, 7036674905, 18331741995, 47757324170, 124416000000},
A—5。{15, 35, 89, 231, 600, 1564, 4084,——30#——A=1,2,3,4,5。

公式。{18, 54, 162, 486, 1446, 4320, 12936, 38724, 115920, 347040, 1038954, 3110400, 9311904, 27877926, 83460792,249864510,748043166,2239488000,6704568348,20072104308,60091768836,179902447026,538591076160,1612431360000}
A—6。{}

northwolves

1. f4[n_]:=Module[{pa,pbc,min=Infinity,best,val,a,b,c},pa=If[Mod[n,4]==0,Table[n/4,4]},PadLeft[#,4]&/@IntegerPartitions[n,3]];
   
2. pbc=Select[Flatten[Permutations/@(PadLeft[#,4]&/@IntegerPartitions[n,4]),1],#[[4]]<=n/2&];
   
3. Do[Do[If[Max[a+b]<=n,Do[If[Max[a+b+c]<=n,val=Total[2^a*3^b*4^c];
   
4. If[val<min,min=val;best={a,b,c}]],{c,pbc}]],{b,pbc}],{a,pa}];{n,best,min}];
   
5. Do[Print@f4[n],{n,16}]

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{1,{{0,0,0,1},{0,0,1,0},{0,1,0,0}},10}
{2,{{0,0,1,1},{0,0,1,1},{1,1,0,0}},20}
{3,{{0,0,0,3},{1,1,1,0},{1,1,1,0}},44}
{4,{{1,1,1,1},{1,1,1,1},{1,1,1,1}},96}
{5,{{0,0,4,1},{1,1,1,2},{2,2,0,1}},216}
{6,{{0,0,5,1},{3,3,0,0},{1,1,1,3}},472}
{7,{{0,0,6,1},{0,5,0,2},{4,0,1,2}},1043}
{8,{{2,2,2,2},{2,2,2,2},{2,2,2,2}},2304}
{9,{{0,7,1,1},{4,2,2,1},{2,0,3,4}},5136}
{10,{{0,8,1,1},{1,1,4,4},{5,1,2,2}},11328}
{11,{{0,9,1,1},{8,1,1,1},{0,1,5,5}},24993}
{12,{{3,3,3,3},{3,3,3,3},{3,3,3,3}},55296}
{13,{{0,11,1,1},{3,0,5,5},{5,2,3,3}},122624}
{14,{{0,12,1,1},{0,0,7,7},{8,2,2,2}},271040}
{15,{{0,10,3,2},{2,2,9,2},{7,2,0,6}},599832}
{16,{{4,4,4,4},{4,4,4,4},{4,4,4,4}},1327104}

northwolves

A—5。{15, 35, 89, 231, 600, 1564, 4084,——30#——A=1,2,3,4,5。
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$n=7,{{{3,0,0,2,3},{3,0,0,2,3},{2,0,3,2,1},{0,4,3,0,1},{0,4,2,2,0}},{900,768,800,810,800},4078}$

northwolves

$n=8: \ {{{3,0,0,2,3},{3,0,0,2,3},{2,0,3,2,1},{0,4,3,0,1},{0,4,2,2,0}},{2000,2000,2160,2160,2304},10624}$

王守恩

northwolves 发表于 2025-10-6 00:04
$n=8: \ {{{3,0,0,2,3},{3,0,0,2,3},{2,0,3,2,1},{0,4,3,0,1},{0,4,2,2,0}},{2000,2000,2160,2160,2304},10 ...

1. (*5组情况：把 5n 个数字 (n个1,n个2,n个3,n个4,n个5) 分成5组，每组n个数字*)
   
2. optimalGrouping5[n_] :=
   
3.   Module[{bestSum = Infinity, bestDist = {}, x1, x2, x3, x4, x5, y1,
   
4.     y2, y3, y4, y5, z1, z2, z3, z4, z5, w1, w2, w3, w4, w5, v1, v2,
   
5.     v3, v4, v5, p1, p2, p3, p4, p5, sum},(*遍历所有有效分配*)
   
6.    For[x1 = 0, x1 <= n, x1++,
   
7.     For[x2 = 0, x2 <= n - x1, x2++,
   
8.      For[x3 = 0, x3 <= n - x1 - x2, x3++,
   
9.       For[x4 = 0, x4 <= n - x1 - x2 - x3, x4++,
   
10.        x5 = n - x1 - x2 - x3 - x4;(*组1:x1个1,x2个2,x3个3,x4个4,x5个5*)
    
11.        For[y1 = 0, y1 <= n - x1, y1++,
    
12.         For[y2 = 0, y2 <= n - x2, y2++,
    
13.          For[y3 = 0, y3 <= n - x3, y3++,
    
14.           For[y4 = 0, y4 <= n - x4, y4++,
    
15.            y5 = n - y1 - y2 - y3 - y4;(*组2*)
    
16.            If[y5 >= 0,
    
17.             For[z1 = 0, z1 <= n - x1 - y1, z1++,
    
18.              For[z2 = 0, z2 <= n - x2 - y2, z2++,
    
19.               For[z3 = 0, z3 <= n - x3 - y3, z3++,
    
20.                For[z4 = 0, z4 <= n - x4 - y4, z4++,
    
21.                 z5 = n - z1 - z2 - z3 - z4;(*组3*)
    
22.                 If[z5 >= 0,
    
23.                  For[w1 = 0, w1 <= n - x1 - y1 - z1, w1++,
    
24.                   For[w2 = 0, w2 <= n - x2 - y2 - z2, w2++,
    
25.                    For[w3 = 0, w3 <= n - x3 - y3 - z3, w3++,
    
26.                     
    
27.                     For[w4 = 0, w4 <= n - x4 - y4 - z4, w4++,
    
28.                     w5 = n - w1 - w2 - w3 - w4;(*组4*)
    
29.                     If[w5 >= 0, v1 = n - x1 - y1 - z1 - w1;(*组5*)
    
30.                     v2 = n - x2 - y2 - z2 - w2;
    
31.                     v3 = n - x3 - y3 - z3 - w3;
    
32.                     v4 = n - x4 - y4 - z4 - w4;
    
33.                     v5 = n - x5 - y5 - z5 - w5;
    
34.                     
    
35.                     If[v1 >= 0 && v2 >= 0 && v3 >= 0 && v4 >= 0 &&
    
36.                     v5 >= 0 && v1 + v2 + v3 + v4 + v5 == n,(*计算5个乘积*)
    
37.                     p1 = 1^x1*2^x2*3^x3*4^x4*5^x5;
    
38.                     p2 = 1^y1*2^y2*3^y3*4^y4*5^y5;
    
39.                     p3 = 1^z1*2^z2*3^z3*4^z4*5^z5;
    
40.                     p4 = 1^w1*2^w2*3^w3*4^w4*5^w5;
    
41.                     p5 = 1^v1*2^v2*3^v3*4^v4*5^v5;
    
42.                     sum = p1 + p2 + p3 + p4 + p5;
    
43.                     If[sum < bestSum, bestSum = sum;
    
44.                     
    
45.                     bestDist = {{x1, x2, x3, x4, x5}, {y1, y2, y3, y4,
    
46.                      y5}, {z1, z2, z3, z4, z5}, {w1, w2, w3, w4,
    
47.                     w5}, {v1, v2, v3, v4, v5}, {p1, p2, p3, p4, p5}};
    
48.                     (*显示重要改进*)
    
49.                     If[bestSum < 10^6,
    
50.                     Print["n=", n, " 新最优解: 和=",
    
51.                     bestSum]];]]]]]]]]]]]]]]]]]]]]];
    
52.    {bestSum, bestDist}];
    
53. 
    
54. (*测试5组情况*)
    
55. Print["=== 测试5组情况 ==="];
    
56. Do[Print["计算 n=", n, "..."];
    
57. result = optimalGrouping5[n];
    
58. Print["n=", n, ": 和=", result[[1]], " 分配=",
    
59.   result[[2]][[1 ;; 5]]];, {n, 2}]
    
60. 
    
61. (*提取数列*)
    
62. results5 = Table[optimalGrouping5[n][[1]], {n, 2}];
    
63. Print["\n5组最优和数列: ", results5];
    
64. Print["一行输出: ", StringJoin[Riffle[ToString /[url=home.php?mod=space&uid=6175]@[/url] results5, ", "]]];

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这个比——OEIS——A260356——还难！

A260356  1, 12, 60, 213, 600, 1443, 3089,  6048,  11041, 19050,  31367,   49658,   76029,    113089,   164031,    232710,    323725,    442510,     595431,         789881,       1034385,      1338715,
公式。  {15, 35, 90, 235, 600, 1565, 4075, 10610, 27640, 72000, 187575, 488660, 1273040, 3316485, 8640000, 22508680, 58638960, 152764520, 397977705, 1036800000, 2701041385, 7036674905, 18331741995, 47757324170, 124416000000},
  A—5。{15, 35, 89, 231, 600, 1564, 4084, 10624, ??????, 72000,——30#——A=1,2,3,4,5。

northwolves

王守恩 发表于 2025-10-6 00:59
明确方向。——我们比A260356还难！
  
公式。  {15, 35, 90, 235, 600, 1565, 4075, 10610, 27640, 7200 ...

$n=9: \ {{{3, 0, 0, 5, 1}, {3, 0, 2, 0, 4}, {1, 3, 3, 0, 2}, {1, 3, 2, 2, 1}, {1, 3, 2, 2, 1}}, {5120, 5625, 5400, 5760, 5760}, 27665}$

northwolves

$n=11:   {{{0, 5, 5, 0, 1}, {0, 6, 2, 3, 0}, {3, 0, 2, 6, 0}, {4, 0, 1, 1, 5}, {4, 0, 1, 1, 5}}, {38880, 36864, 36864, 37500, 37500}, 187608}$

northwolves

{12,{{0, 7, 1, 4, 0}, {1, 4, 5, 0, 2}, {3, 0, 5, 2, 2}, {4, 0, 1, 4, 3}, {4, 1, 0, 2, 5}}, {98304, 97200, 97200, 96000, 100000}, 488704}
{13,{{0, 6, 5, 2, 0}, {1, 5, 4, 1, 2}, {4, 0, 4, 0, 5}, {4, 1, 0, 5, 3}, {4, 1, 0, 5, 3}}, {248832, 259200, 253125, 256000, 256000}, 1273157}
{14,{{0,9,0,4,1},{2,2,8,0,2},{4,1,0,8,1},{4,1,3,1,5},{4,1,3,1,5}},{655360,656100,655360,675000,675000},3316820}
{15,{{3,3,3,3,3},{3,3,3,3,3},{3,3,3,3,3},{3,3,3,3,3},{3,3,3,3,3}},8640000}

王守恩

把 A—2 补一补。

w(1)=3, 3=1+2,
1=1,
2=2,

w(2)=4,  4=2+2,
1×2=2,
2×1=2,

w(3)=6,  6=2+4,
1×2×1=2,
2×1×2=4,

w(4)=8,  8=4+4,
1×2×1×2=4,
2×1×2×1=4,

w(5)=12,  12=4+8,
1×2×1×2×1=4,
2×1×2×1×2=8,

w(6)=16,  16=8+8,
1×2×1×2×1×2=8,
2×1×2×1×2×1=8,

得到一串数—{3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, —— w(n) = (4 + 3 Sqrt[2] + (4 - 3 Sqrt[2]) Cos[n*Pi])*2^(n/2 - 2) 。

王守恩

继续出题！——看似简单。规律不好找。

w(5)=15,  
1=1,
2=2,
3=3,
4=4,
5=5,

w(6)=20,  
1×2=2,
3=3,
4=4,
5=5,
6=6,

w(7)=28,  
1×4=4,
2×3=6,
5=5,
6=6,
7=7,

w(8)=43,  
1×6=6,
2×5=10,
3×4=12,
7=7,
8=8,

w(9)=69,  
1×8=8,
2×7=14,
3×6=18,
4×5=20,
9=9,

w(10)=110,  
1×10=10,
2×9=18,
3×8=24,
4×7=28,
5×6=30,

w(11)=170,  
1×2×11=22,
3×10=30,
4×9=36,
5×8=40,
6×7=42,

w(12)=273,  
1×4×12=48,
2×3×10=60,
5×11=55,
6×9=54,
7×8=56,

w(13)=455,  
1×7×13=91,
2×4×12=96,
3×5×6=90,
8×11=88,
9×10=90,

w(14)=772,  
1×11×14=154,
2×8×9=144,
3×5×10=150,
4×6×7=168,
12×13=156,

w(15)=1336,  ——有问题了。
1×14×15=210,
2×11×13=286,
3×12×7=252,
4×8×9=288,
5×6×10=300,