三重内积(增加难度)的最小值

王守恩 · 发表于 2025-10-1 07:59:47

楼上是这样一串数——2, 6, 13, 22, 36, 53, 75, 102, 135, 173, 219, 271, 331, 399, 476, 561, 657, 762, 878, 1005, 1144, 1294, 1458, 1634, 1824, 2028, 2247, 2480, 2730, 2995, ......

Table[Round[(26 n + 19 n^2 + 2 n^3)/24], {n, 40}]——通项公式出奇的好——OEIS是没有的——这串数也是有现实意义。

n=1: a={2} b={1} 和=2= 2(a)*1(b)*1(c)
n=2: a={1,2} b={1,2} 和=6= 1(a)*1(b)*2(c) + 2(a)*2(b)*1(c)
n=3: a={1,2,3} b={1,1,2} 和=13= 1(a)*1(b)*3(c) + 2(a)*1(b)*2(c) + 3(a)*2(b)*1(c)
n=4: a={1,1,2,3} b={1,2,3,1} 和=22=1(a)*1(b)*4(c) + 1(a)*2(b)*3(c) + 2(a)*3(b)*1(c) + 3(a)*1(b)*2(c)
n=5: a={1,1,2,3,4} b={1,2,1,3,1} 和=36= 1(a)*1(b)*5(c) + 1(a)*2(b)*4(c) + 2(a)*1(b)*3(c) + 3(a)*3(b)*1(c) + 4(a)*1(b)*2(c)
n=6: a={1,1,1,2,3,4} b={1,2,3,1,4,1} 和=53=1(a)*1(b)*6(c) + 1(a)*2(b)*5(c) + 1(a)*3(b)*3(c) + 2(a)*1(b)*4(c) + 3(a)*4(b)*1(c) + 4(a)*1(b)*2(c)
n=7: a={1,1,1,2,3,4,5} b={1,2,3,1,4,1,1} 和=75
n=8: a={1,1,1,1,2,3,4,5} b={1,2,3,4,1,5,1,1} 和=102
n=9: a={1,1,1,1,2,3,4,5,6} b={1,2,3,4,1,1,5,1,1} 和=135
n=10: a={1,1,1,1,1,2,3,4,5,6} b={1,2,3,4,6,1,1,5,1,1} 和=173
n=11: a={1,1,1,1,1,2,3,4,5,6,7} b={1,2,3,4,5,1,1,6,1,1,1} 和=219
n=12: a={1,1,1,1,1,1,2,3,4,5,6,7} b={1,2,3,4,5,7,1,1,1,6,1,1} 和=271
n=13: a={1,1,1,1,1,1,2,3,4,5,6,7,8} b={1,2,3,4,5,7,1,1,1,6,1,1,1} 和=331
n=14: a={1,1,1,1,1,1,1,2,3,4,5,6,7,8} b={1,2,3,4,5,6,8,1,1,1,7,1,1,1} 和=399
n=15: a={1,1,1,1,1,1,1,2,3,4,5,6,7,8,9} b={1,2,3,4,5,6,8,1,1,1,1,7,1,1,1} 和=476

SystemException["MemoryAllocationFailure"]——说来不了了。

楼上的解法是来不了了——有一个想法——若能把a与b合并为一个数字串——b是关键——什么规律？谢谢！

王守恩 · 发表于 2025-10-1 11:28:38

还是简单一点。

Table[a = Join[ConstantArray[1, Floor[(n + 1)/2]], Range[2, n - Floor[(n + 1)/2] + 1]];
b = Join[ConstantArray[1, Floor[n/2]], Range[n - Floor[n/2] + 1 + Mod[n - 1, 2], n + 1]];
k = MinimalBy[DeleteDuplicates[Permutations[b]], Total[ReverseSort[a*#]*Range[n]] &] // First;
m = Total[ReverseSort[a*k]*Range[n]];
Print["n=", n, ": a=", a, " b=", k, " 和=", m], {n, 20}]

复制代码

n=1: a={1} b={2} 和=2
n=2: a={1,2} b={3,1} 和=7
n=3: a={1,1,2} b={3,4,1} 和=16
n=4: a={1,1,2,3} b={4,5,1,1} 和=30
n=5: a={1,1,1,2,3} b={4,5,6,1,1} 和=50
n=6: a={1,1,1,2,3,4} b={5,6,7,1,1,1} 和=77
n=7: a={1,1,1,1,2,3,4} b={5,6,7,8,1,1,1} 和=112
n=8: a={1,1,1,1,2,3,4,5} b={6,7,8,9,1,1,1,1} 和=156
n=9: a={1,1,1,1,1,2,3,4,5} b={6,7,8,9,10,1,1,1,1} 和=210
n=10: a={1,1,1,1,1,2,3,4,5,6} b={7,8,9,10,11,1,1,1,1,1} 和=275
n=11: a={1,1,1,1,1,1,2,3,4,5,6} b={7,8,9,10,11,12,1,1,1,1,1} 和=352
n=12: a={1,1,1,1,1,1,2,3,4,5,6,7} b={8,9,10,11,12,13,1,1,1,1,1,1} 和=442
n=13: a={1,1,1,1,1,1,1,2,3,4,5,6,7} b={8,9,10,11,12,13,14,1,1,1,1,1,1} 和=546

SystemException["MemoryAllocationFailure"]——说来不了了。

{2, 7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442, 546, 665, 800, 952, 1122, 1311, 1520, 1750, 2002, 2277, 2576, 2900, 3250, 3627, 4032, 4466, 4930, 5425, 5952, 6512, 7106, 7735, 8400, 9102, 9842, 10621, 11440, 12300}

通项公式可以这样—a(n) = Total[Range[2, n + 1]*Reverse[Range[n]]] 。通项公式也可以这样—a(n) = (n^3 + 6 n^2 + 5 n)/6 。A005581 — a(n) = (n-1)*n*(n+4)/6.——没有这样的内容。

王守恩 · 发表于 2025-10-4 05:55:22

试试！走自己的路！

w(1)=6, 6=1+2+3,
1=1,
2=2,
3=3,

w(2)=10,  10=3+4+3,
1×3=3,
2×2=4,
3×1=3,

w(3)=18,  18=6+6+6,
1×2×3=6,
2×3×1=6,
3×1×2=6,

w(4)=33,  33=9+12+12,
1×1×3×3=9,
2×3×1×2=12,
3×2×2×1=12,

w(5)=60,  60=24+18+18,
1×2×2×3×2=24,
2×3×3×1×1=18,
3×1×1×2×3=18,

w(6)=108,  108=36+36+36,
1×1×2×2×3×3=36,
2×2×3×3×1×1=36,
3×3×1×1×2×2=36,

w(7)=198,  198=54+72+72,
1×1×1×2×3×3×3=54,
2×2×3×3×1×1×2=72,
3×3×2×1×2×2×1=72,

w(8)=360,  360=108+144+108,
1×1×1×2×2×3×3×3=108,
2×2×2×3×3×1×1×2=144,
3×3×3×1×1×2×2×1=108,

w(9)=648,  648=216+216+216,
1×1×1×2×2×2×3×3×3=216,
2×2×2×3×3×3×1×1×1=216,
3×3×3×1×1×1×2×2×2=216,

得到一串数——6, 10, 18, 33, 60, 108, 198, 360, 648, 1188? —— 怎么走？

northwolves · 发表于 2025-10-4 09:15:08

显然有 a(n+3)=6a(n)

northwolves · 发表于 2025-10-4 09:54:17

Table[Round[(37+18*Cos[2*(n+1)*Pi/3]-10*Cos[2*n*Pi/3])*6^Floor[n/3-1]],{n,50}]

复制代码

{6,10,18,33,60,108,198,360,648,1188,2160,3888,7128,12960,23328,42768,77760,139968,256608,466560,839808,1539648,2799360,5038848,9237888,16796160,30233088,55427328,100776960,181398528,332563968,604661760,1088391168,1995383808,3627970560,6530347008,11972302848,21767823360,39182082048,71833817088,130606940160,235092492288,431002902528,783641640960,1410554953728,2586017415168,4701849845760,8463329722368,15516104491008,28211099074560}

王守恩 · 发表于 2025-10-4 10:44:58

northwolves 发表于 2025-10-4 09:54
{6,10,18,33,60,108,198,360,648,1188,2160,3888,7128,12960,23328,42768,77760,139968,256608,466560,83 ...

{6,10,18,33,60,108,198,360,648,1188,2145,3888,7083,12844,23328,42498,77064,139968,254988,462384,839808,1526769,2774304,5038848,9160614,16645824,30233088,54963684,99874944,181398528,329782104,599249664,1088391168}

王守恩 · 发表于 2025-10-4 14:46:01

northwolves 发表于 2025-10-4 09:15
显然有 a(n+3)=6a(n)

试试！走自己的路！

w(1)=6, 6=1+2+3,
1=1,
2=2,
3=3,

w(2)=10,  10=3+4+3,
1×3=3,
2×2=4,
3×1=3,

w(3)=18,  18=6+6+6,
1×2×3=6,
2×3×1=6,
3×1×2=6,

w(4)=33,  33=9+12+12,
1×1×3×3=9,
2×3×1×2=12,
3×2×2×1=12,

w(5)=60,  60=24+18+18,
1×2×2×3×2=24,
2×3×3×1×1=18,
3×1×1×2×3=18,

w(6)=108,  108=36+36+36,
1×1×2×2×3×3=36,
2×2×3×3×1×1=36,
3×3×1×1×2×2=36,

w(7)=198,  198=54+72+72,
1×1×1×2×3×3×3=54,
2×2×3×3×1×1×2=72,
3×3×2×1×2×2×1=72,

w(8)=360,  360=108+144+108,
1×1×1×2×2×3×3×3=108,
2×2×2×3×3×1×1×2=144,
3×3×3×1×1×2×2×1=108,

w(9)=648,  648=216+216+216,
1×1×1×2×2×2×3×3×3=216,
2×2×2×3×3×3×1×1×1=216,
3×3×3×1×1×1×2×2×2=216,

w(10)=1188,  1188=432+432+324,
1×1×1×2×2×2×3×3×3×2=432,—{3,4,3},
2×2×2×3×3×3×2×1×1×1=432,—{3,4,3},
3×3×3×1×1×1×1×2×2×3=324,—{4,2,4},

w(11)=2145,  2145=768+648+729,
1×1×2×2×2×2×3×2×2×2×2=768,—{2,8,1},
2×2×3×3×3×3×2×1×1×1×1=648,—{4,3,4},
3×3×1×1×1×1×1×3×3×3×3=729,—{5,0,6},

w(12)=3888,  3888=1296+1296+1296,
1×1×1×1×2×2×2×2×3×3×3×3=1296,—{4,4,4},
2×2×2×2×3×3×3×3×1×1×1×1=1296,—{4,4,4},
3×3×3×3×1×1×1×1×2×2×2×2=1296,—{4,4,4},

w(13)=7083,  7083=2304+2592+2187,
1×1×1×2×2×2×2×2×2×2×2×3×3=2304,—{3,8,2},
2×2×2×3×3×3×1×1×1×1×3×2×2=2592,—{4,5,4},
3×3×3×1×1×1×3×3×3×3×1×1×1=2187,—{6,0,7},

w(14)=12844,  12844=4096+4374+4374,
1×1×2×2×2×2×2×2×2×2×2×2×2×2=4096,—{2,12,0},
2×3×1×1×1×1×1×1×3×3×3×3×3×3=4374,—{6,1,7},
3×2×3×3×3×3×3×3×1×1×1×1×1×1=4374,—{6,1,7},

w(15)=23328,——{5,5,5},{5,5,5},{5,5,5},
w(16)=42498, ——{4,9,3},{5,6,5},{7,1,8},
w(17)=77064, ——{3,13,1},{7,2,8},{7,2,8},
w(18)=139968, ——{6,6,6},{6,6,6},{6,6,6},
w(19)=254988, ——{5,10,4},{6,7,6},{8,2,9},
w(20)=462384, ——{4,14,2},{8,3,9},{8,3,9},
w(21)=839808, ——{7,7,7},{7,7,7},{7,7,7},

得到一串数——{6, 10, 18, 33, 60, 108, 198, 360, 648, 1188, 2145, 3888, 7083, 12844, 23328, 42498, 77064, 139968, 254988, 462384}——不可以有通项公式。

王守恩 · 发表于 2025-10-4 17:51:14

试试！走自己的路！

w(1)=10,  10=1+2+3+4,
1=1,
2=2,
3=3,
4=4,

w(2)=20,  20=4+6+6+4,
1×4=4,
2×3=6,
3×2=6,
4×1=4,

w(3)=44,  44=12+12+12+8,
1×3×4=12,
2×2×3=12,
3×4×1=12,
4×1×2=8,

w(4)=96,  96=24+24+24+24,
1×2×3×4=24,—{1,1,1,1},
2×1×2×3=24,—{1,1,1,1},
3×4×4×1=24,—{1,1,1,1},
4×3×1×2=24,—{1,1,1,1},

w(5)=214,  214=54+64+48+48,
1×2×3×3×3=54,—{1,1,3,0}
2×4×2×2×2=64,—{0,4,0,1},
3×1×4×4×1=48,—{2,0,1,2},
4×3×1×1×4=48,—{2,0,1,2},

w(6)=472,  472=128+128+108+108,
1×1×2×4×4×4=128,—{2,1,0,3},
2×2×4×2×2×2=128,—{0,5,0,1},
3×4×1×1×3×3=108,—{2,0,3,1},
4×3×3×3×1×1=108,—{2,0,3,1},

w(7)=1043,  1043=256+288+243+256,
1×1×1×4×4×4×4=256,—{3,0,0,4},
2×2×2×2×2×3×3=288,—{0,5,2,0},
3×3×3×3×3×1×1=243,—{2,0,5,0},
4×4×4×1×1×2×2=256,—{2,2,0,3},

w(8)=2304,  2304=576+576+576+576,
1×1×2×2×3×3×4×4=576,—{2,2,2,2},
2×2×3×3×2×2×2×2=576,—{0,6,2,0},
3×4×4×4×1×1×1×3=576,—{3,0,2,3},
4×3×1×1×4×4×3×1=576,—{3,0,2,3},

w(9)=5136,  5136=1152+1536+1152+1296,
1×1×1×2×3×3×4×4×4=1152,—{3,1,2,3},
2×2×2×4×2×2×2×2×3=1536,—{0,7,1,1},
3×4×4×1×1×4×1×3×2=1152,—{3,1,2,3},
4×3×3×3×4×1×3×1×1=1296,—{3,0,4,2},

w(10)=11328, ——{0,8,1,1},{3,1,4,2},{3,1,4,2},{4,0,1,5},
w(11)=24993, ——{0,9,1,1},{3,0,8,0},{4,1,1,5},{4,1,1,5},
w(12)=55296, ——{0,9,3,0},{4,1,3,4},{4,1,3,4},{4,1,3,4},

得到一串数——10, 20, 44, 96, 214, 472, 1043, 2304, 5136, 11328, 24993, 55296, 122624,——不可以有通项公式。

补充内容 (2025-10-5 15:20):
补充1项——271040。——别指望我。我已经很努力了。

northwolves · 发表于 2025-10-4 21:25:37

王守恩发表于 2025-10-4 14:46
试试！走自己的路！

w(1)=6, 6=1+2+3,

f[n_]:=(x=Select[Flatten[Table[Table[{a,r[[2]]},{a,Permutations[r[[1]]]}],{r,Tuples[PadLeft@IntegerPartitions[n,3],{2}]}],1],Max[#[[1]]+#[[2]]]<=n&];{n,SortBy[Table[{r,Total[2^r[[1]]*3^r[[2]]]},{r,x}],Last][[1]]});Do[Print@f[n],{n,2,40}]

复制代码

{2,{{{1,1},{1,1}},12}}
{3,{{{1,1,1},{1,1,1}},18}}
{4,{{{0,2,2},{2,1,1}},33}}
{5,{{{1,1,3},{2,2,1}},60}}
{6,{{{2,2,2},{2,2,2}},108}}
{7,{{{1,3,3},{3,2,2}},198}}
{8,{{{2,2,4},{3,3,2}},360}}
{9,{{{3,3,3},{3,3,3}},648}}
{10,{{{2,4,4},{4,3,3}},1188}}
{11,{{{0,3,8},{6,4,1}},2145}}
{12,{{{4,4,4},{4,4,4}},3888}}
{13,{{{0,5,8},{7,4,2}},7083}}
{14,{{{12,1,1},{0,7,7}},12844}}
{15,{{{5,5,5},{5,5,5}},23328}}
{16,{{{1,6,9},{8,5,3}},42498}}
{17,{{{2,2,13},{8,8,1}},77064}}
{18,{{{6,6,6},{6,6,6}},139968}}
{19,{{{2,7,10},{9,6,4}},254988}}
{20,{{{3,3,14},{9,9,2}},462384}}
{21,{{{7,7,7},{7,7,7}},839808}}
{22,{{{0,11,11},{12,5,5}},1526769}}
{23,{{{4,4,15},{10,10,3}},2774304}}
{24,{{{8,8,8},{8,8,8}},5038848}}
{25,{{{1,12,12},{13,6,6}},9160614}}
{26,{{{5,5,16},{11,11,4}},16645824}}
{27,{{{9,9,9},{9,9,9}},30233088}}
{28,{{{2,13,13},{14,7,7}},54963684}}
{29,{{{6,6,17},{12,12,5}},99874944}}
{30,{{{10,10,10},{10,10,10}},181398528}}
{31,{{{3,14,14},{15,8,8}},329782104}}
{32,{{{7,7,18},{13,13,6}},599249664}}
{33,{{{11,11,11},{11,11,11}},1088391168}}
{34,{{{4,15,15},{16,9,9}},1978692624}}
{35,{{{0,8,27},{19,14,2}},3594661083}}
{36,{{{12,12,12},{12,12,12}},6530347008}}
{37,{{{5,16,16},{17,10,10}},11872155744}}
{38,{{{1,9,28},{20,15,3}},21567966498}}
{39,{{{13,13,13},{13,13,13}},39182082048}}
{40,{{{6,17,17},{18,11,11}},71232934464}}

王守恩 · 发表于 2025-10-5 10:36:45

继续出题！

w(1)=15,
1=1,
2=2,
3=3,——第1列是固定的。
4=4,
5=5,

w(2)=35,
1×5=5,
2×4=8,
3×3=9,——第1列是固定的。
4×2=8,
5×1=5,

w(3)=89,
1×3×5=15,——第1行是最小的。
2×5×2=20,
3×2×3=18,——剩下的机会不多。
4×1×4=16,
5×4×1=20,

w(4)=96,
1×2×4×5=40,——第1行是最小的。
2×4×3×2=48,
3×1×5×3=45,——剩下的机会不多。
4×3×1×4=48,
5×5×2×1=50,

w(5)=600,
1×2×3×4×5=120,——第1行是最小的。
2×3×4×5×1=120,
3×4×5×1×2=120,——剩下的机会不多。
4×5×1×2×3=120,
5×1×2×3×4=120,

w(6)=1564,
1×1×3×4×5×5=300,——第1行是最小的。
2×2×5×2×4×2=320,
3×3×2×3×2×3=324,——剩下的机会不多。
4×4×4×5×1×1=320,
5×5×1×1×3×4=300,

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[原创] 三重内积(增加难度)的最小值

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