数字串的通项公式

王守恩 · 发表于 2023-7-5 14:37:40

宽(AB)为n(正整数)的长方形(长=宽*\(\sqrt{2}\))内动点P，三角形ABP三边长为整数，问动点P可能有几个？

得到一串数(OEIS好像没有):1,4,11,18,27,40,53,70,89,110,133,160,185,214,249,280,321,356,397,440,...

Table[Dimensions@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b],(m*n)/(k*k)≤Sqrt[2]/Sin[a+b],\[Pi]/2≥a>0,\[Pi]/2≥b>0},{a,b}],{m,1,Sqrt[3]k},{n,1,Sqrt[3]k}]]/2,{k,1,20}]

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算式没问题，就是太慢了。

northwolves · 发表于 2023-7-5 14:50:01

王守恩发表于 2023-7-5 14:16
上底(AB)为n(正整数)的等腰梯形(底角=60,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个？

...

没看明白。你画一下n=3的情况

northwolves · 发表于 2023-7-5 14:55:48

王守恩发表于 2023-7-5 09:07
边长为n(正整数)的正三角形ABC内动点P，三角形ABP三边长为整数，问动点P可能有几个？

得到一串数(OEIS好 ...

这个数据有误。a(1)=0,a(2)=0,a(3)=1(3,2,2)

王守恩 · 发表于 2023-7-5 15:46:21

本帖最后由王守恩于 2023-7-5 15:50 编辑

northwolves 发表于 2023-7-5 14:55
这个数据有误。a(1)=0,a(2)=0,a(3)=1(3,2,2)

a(1)={1,1},
a(2)={2,2},
a(3)={2,2},{3,3},
a(4)={2,3},{3,2},{3,3},{4,4},
a(5)={2,4},{3,3},{3,4},{4,2},{4,3},{4,4},{5,5},

王守恩 · 发表于 2023-7-5 16:22:59

本帖最后由王守恩于 2023-7-5 16:27 编辑

上底(AB)为n(正整数)的等腰梯形(底角=60,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

得到一串数(OEIS好像没有):1,5,12,20,31,43,62,78,98,124,147,175,206,244,275,315,355,395,442,488,...

Table[Dimensions@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b],(m*n)/(k*k)≤Sin[\[Pi]/2]/Sin[a+b],2\[Pi]/3≥a>0,2\[Pi]/3≥b>0},{a,b}],{m,1,(k Sqrt[21+6Sqrt[3]])/3},{n,1,(k Sqrt[21+6Sqrt[3]])/3}]]/2,{k,1,20}]

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a(1)={1,1},
a(2)={1,2},{2,1},{2,2},{2,3},{3,2},
a(3)={1,3},{2,2},{2,3},{2,4},{3,1},{3,2},{3,3},{3,4},{3,5},{4,2},{4,3},{5,3},

northwolves · 发表于 2023-7-5 19:46:03

王守恩发表于 2023-7-5 16:22
上底(AB)为n(正整数)的等腰梯形(底角=60,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

...

我算的 a(7)=70:

{{1,7},{2,6},{2,7},{2,8},{3,5},{3,6},{3,7},{3,8},{4,4},{4,5},{4,6},{4,7},{4,8},{4,9},{5,3},{5,4},{5,5},{5,6},{5,7},{5,8},{5,9},{5,10},{6,2},{6,3},{6,4},{6,5},{6,6},{6,7},{6,8},{6,9},{6,10},{6,11},{7,1},{7,2},{7,3},{7,4},{7,5},{7,6},{7,7},{7,8},{7,9},{7,10},{7,11},{7,12},{8,2},{8,3},{8,4},{8,5},{8,6},{8,7},{8,9},{8,10},{8,11},{8,12},{8,13},{9,4},{9,5},{9,6},{9,7},{9,8},{10,5},{10,6},{10,7},{10,8},{11,6},{11,7},{11,8},{12,7},{12,8},{13,8}}

王守恩 · 发表于 2023-7-6 07:26:46

northwolves 发表于 2023-7-5 19:46
我算的 a(7)=70:

{{1,7},{2,6},{2,7},{2,8},{3,5},{3,6},{3,7},{3,8},{4,4},{4,5},{4,6},{4,7},{4,8}, ...

太高(高=上底)了: {8,9},{8,10},{8,11},{8,12}{9,8},{10,8},{11,8},{12,8},

王守恩 · 发表于 2023-7-6 11:48:26

上底(AB)为n(正整数)的等腰梯形(底角=45,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

得到一串数(OEIS好像没有):{1, 5, 12, 22, 37, 53, 70, 88, 114, 142, 173, 205, 240, 280, 319, 369, 417, 463, 514, 568}

Table[Length@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b]≥(m*n)/k,3\[Pi]/4≥a>0,3\[Pi]/4≥b>0},{a,b}],{m,1,Sqrt[5]k},{n,1,Sqrt[5]k}]]/2,{k,1,20}]

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算式没问题，就是太慢了。

northwolves · 发表于 2023-7-6 23:36:44

王守恩发表于 2023-7-6 11:48
上底(AB)为n(正整数)的等腰梯形(底角=45,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

...

Table[Floor[Sqrt@5n/2]-Floor[n/2]+2Length@Select[Subsets[Range[Sqrt@5n],{2}],Total@#>n&&-Sqrt@2#[[1]]n<#[[1]]^2+n^2-#[[2]]^2&&(#[[2]]^2-#[[1]]^2)^2>n^2*(2Total@(#^2)-5n^2)&],{n,50}]

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{1,5,12,22,37,53,70,88,114,142,173,205,240,280,319,369,417,463,514,568,631,689,758,822,891,964,1041,1125,1202,1288,1373,1465,1562,1655,1756,1854,1959,2065,2184,2292,2407,2531,2649,2775,2908,3036,3169,3307,3450,3588}

王守恩 · 发表于 2023-7-7 13:12:31

1,底(AB)为n(正整数)的等腰三角形(底角=15)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a] == n/Sin[b] == k/Sin[a + b] >= (2 m n)/(k Tan[15 \[Pi]/180] ), 15 \[Pi]/180 >= a > 0, 15 \[Pi]/180 >= b>0},{a,b}],{m,1,k},{n,1,k}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,}

2,底(AB)为n(正整数)的等腰三角形(底角=30)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a] == n/Sin[b] == k/Sin[a + b] >= (2 m n)/(k Tan[30 \[Pi]/180] ), 30 \[Pi]/180 >= a > 0, 30 \[Pi]/180 >=b>0},{a, b}],{m,1,k},{n,1,k}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 5, 7, 7, 9, 11, 13, 15,}

3,底(AB)为n(正整数)的等腰三角形(底角=45)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a] == n/Sin[b] == k/Sin[a + b] >= (2 m n)/(k Tan[45 \[Pi]/180] ), 45 \[Pi]/180 >= a > 0, 45 \[Pi]/180 >=b>0},{a,b}],{m,1,k},{n,1,k}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{0, 0, 1, 0, 1, 3, 5, 5, 8, 12, 12, 16, 21, 24, 25, 31, 38, 39, 44, 52,}

4,底(AB)为n(正整数)的等腰三角形(底角=60)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a] == n/Sin[b] == k/Sin[a + b] >= (2 m n)/(k Tan[60 \[Pi]/180] ), 60 \[Pi]/180 >= a > 0, 60 \[Pi]/180 >=b>0},{a,b}],{m,1,k},{n,1,k}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有): {1, 1, 2, 4, 7, 11, 16, 20, 21, 27, 34, 42, 51, 61, 72, 78, 85, 97, 110, 124,}

5,底(AB)为n(正整数)的等腰三角形(底角=75)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a] == n/Sin[b] == k/Sin[a + b] >= (2 m n)/( k Tan[75 \[Pi]/180]), 75 \[Pi]/180 >= a > 0, 75 \[Pi]/180 >=b>0},{a,b}], {m,1,k},{n,1,k}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{1, 1, 4, 6, 11, 15, 22, 28, 37, 45, 56, 66, 79, 91, 106, 120, 137, 153, 172, 190,}

6,边长为n(正整数)的正方形ABCD内动点P，三角形ABP三边长为整数，问动点P可能有几个？

Table[Length@Flatten[Table[NSolve[{m/Sin[a] == n/Sin[b] == k/Sin[a + b] >= (m n)/k, 90 \[Pi]/180 >= a > 0, 90 \[Pi]/180 >= b > 0}, {a, b}], {m, 1, k Sqrt[2]}, {n, 1, k Sqrt[2]}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{1, 3, 8, 14, 21, 29, 40, 52, 66, 82, 99, 117, 138, 160, 185, 209, 239, 265, 294, 326,}

7,上底(AB)为n(正整数)的等腰梯形(底角=75,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b]≥(m n)/k,105\[Pi]/180≥a>0,105\[Pi]/180≥b>0},{a,b}],{m,1,k Sqrt[13-6Sqrt[3]]},{n,1,k Sqrt[13-6Sqrt[3]]}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{1, 5, 10, 16, 25, 37, 50, 64, 84, 102, 123, 149, 174, 202, 231, 263, 295, 331, 368, 406,}

8,上底(AB)为n(正整数)的等腰梯形(底角=60,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b]≥(m n)/k,120\[Pi]/180≥a>0,120\[Pi]/180≥b>0},{a,b}],{m,1,k Sqrt[(7+2Sqrt[3])/3]},{n,1,k Sqrt[(7+2Sqrt[3])/3]}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{1, 5, 12, 20, 31, 43, 62, 78, 98, 124, 147, 175, 206, 244, 275, 315, 355, 395, 442, 488,}

9,上底(AB)为n(正整数)的等腰梯形(底角=45,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a] == n/Sin[b] == k/Sin[a + b] >= (m n)/k,135\[Pi]/180>=a>0,135\[Pi]/180>=b>0},{a,b}],{m,1,Sqrt[5]k},{n,1,Sqrt[5]k}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{1, 5, 12, 22, 37, 53, 70, 88, 114, 142, 173, 205, 240, 280, 319, 369, 417, 463, 514, 568,}

10,上底(AB)为n(正整数)的等腰梯形(底角=30,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b]≥(m n)/k,150\[Pi]/180≥a>0,150\[Pi]/180≥b>0},{a,b}],{m,1,k Sqrt[5+2Sqrt[3]]},{n,1,k Sqrt[5+2Sqrt[3]]}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{1, 5, 12, 22, 37, 55, 80, 104, 136, 170, 207, 239, 276, 322, 365, 419, 473, 533, 594, 660,}

11,上底(AB)为n(正整数)的等腰梯形(底角=15,上底=高)内动点P，三角形ABP三边长为整数，问动点P可能有几个?

Table[Length@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b]≥(mn)/k,165\[Pi]/180≥a>0,165\[Pi]/180≥b>0},{a,b}],{m,1,kSqrt[13+6Sqrt[3]]},{n,1,kSqrt[13+6Sqrt[3]]}]]/2,{k,1,20}]

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得到一串数(OEIS好像没有):{1, 5, 12, 22, 37, 55, 80, 104, 136, 170, 207, 253, 300, 354, 407, 471, 535, 603, 674, 748,}

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[原创] 数字串的通项公式

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