数字串的通项公式

northwolves · 发表于 2023-7-7 13:35:46

本帖最后由 northwolves 于 2023-7-7 13:38 编辑

王守恩发表于 2023-7-7 13:12
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?

得到一 ...

6-11题：

Table[{di,Table[Floor[Sqrt@5n/2]-Floor[n/2]+2Length@Select[Subsets[Range[n*Sqrt[1+1/(Sin[di*Pi/180])]^2],{2}],Total@#>n&&#[[1]]^2+n^2+2Cos[di*Pi/180]#[[1]]n>=#[[2]]^2 &&(#[[2]]^2-#[[1]]^2)^2>=n^2*(2Total@(#^2)-5n^2)&],{n,20}]},{di,15,90,15}]// MatrixForm

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$
\begin{array}{cc}
15 & \{1,5,12,22,37,55,80,104,136,170,207,253,300,354,407,471,535,603,674,748\} \\
30 & \{1,5,12,22,37,55,80,104,136,170,207,239,276,322,365,419,473,533,594,660\} \\
45 & \{1,5,12,22,37,53,70,88,114,142,173,205,240,280,319,369,417,463,514,568\} \\
60 & \{1,5,12,20,31,43,62,78,98,124,147,175,206,244,275,315,355,395,442,488\} \\
75 & \{1,5,10,16,25,37,50,64,84,102,123,149,174,202,231,263,295,331,368,406\} \\
90 & \{1,3,8,14,21,29,40,52,66,82,99,117,138,160,185,209,239,265,294,326\} \\
\end{array}$

northwolves · 发表于 2023-7-7 13:52:56

本帖最后由 northwolves 于 2023-7-7 13:55 编辑

梯形内的点应该去掉等号：

Table[{di,Table[Floor[Sqrt@5n/2]-Floor[n/2]+2Length@Select[Subsets[Range[n*Sqrt[1+1/(Sin[di*Pi/180])]^2],{2}],Total@#>n&&#[[1]]^2+n^2+2Cos[di*Pi/180]#[[1]]n>#[[2]]^2 &&(#[[2]]^2-#[[1]]^2)^2>n^2*(2Total@(#^2)-5n^2)&],{n,20}]},{di,15,90,15}]

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{{15,{1,5,12,22,37,55,80,104,136,170,207,253,300,354,407,471,535,603,674,748}},
{30,{1,5,12,22,37,55,80,104,136,170,207,239,276,322,365,419,473,533,594,660}},
{45,{1,5,12,22,37,53,70,88,114,142,173,205,240,280,319,369,417,463,514,568}},
{60,{1,5,12,20,29,43,60,76,98,122,147,175,206,242,273,311,355,395,442,486}},
{75,{1,5,10,16,25,37,50,64,84,102,123,149,174,202,231,263,295,331,368,406}},
{90,{1,3,8,12,21,29,40,50,66,82,99,113,138,160,183,207,239,265,294,324}}}

王守恩 · 发表于 2023-7-7 19:43:20

谢谢 northwolves ! 6—11这样就出来了！谢谢 northwolves ！这些按钮我不会用。

Table[2Length@Select[Subsets[Range[5n],{2}],Total@#>n&&#[[1]]^2+n^2+2Sin[k*Pi/12]#[[1]]n≥#[[2]]^2&&(#[[2]]^2-#[[1]]^2)^2≥n^2(2Total@#^2-5n^2)&]+Floor[Sqrt@5n/2]-Floor[n/2],{k,0,5},{n,20}]

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{1, 3, 8, 14, 21, 29, 40, 52, 66, 82, 99, 117, 138, 160, 185, 209, 239, 265, 294, 326},
{1, 5, 10, 16, 25, 37, 50, 64, 84, 102, 123, 149, 174, 202, 231, 263, 295, 331, 368, 406},
{1, 5, 12, 20, 31, 43, 62, 78, 98, 124, 147, 175, 206, 244, 275, 315, 355, 395, 442, 488},
{1, 5, 12, 22, 37, 53, 70, 88, 114, 142, 173, 205, 240, 280, 319, 369, 417, 463, 514, 568},
{1, 5, 12, 22, 37, 55, 80, 104, 136, 170, 207, 239, 276, 322, 365, 419, 473, 533, 594, 660},
{1, 5, 12, 22, 37, 55, 80, 104, 136, 170, 207, 253, 300, 354, 407, 471, 535, 603, 674, 748}

northwolves · 发表于 2023-7-7 23:59:40

王守恩发表于 2023-7-7 13:12
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?

得到一 ...

1-5题：

Table[{di,Table[Floor[n Sec[di Degree]/2]-Floor[n/2]+2Length@Select[Subsets[Range[n*Sqrt[1+Csc[di Degree]]^2],{2}],Total@#>n&&#[[1]]^2+n^2>#[[2]]^2+2Cos[di Degree]#[[1]]n&&#[[2]]^2+n^2<#[[1]]^2+2n^2#[[2]]&],{n,20}]},{di,15,75,15}]

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{{15,{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}},
{30,{0,0,0,0,0,0,1,0,1,2,3,4,5,5,7,7,9,11,13,15}},
{45,{0,0,1,0,1,3,5,5,8,12,12,16,21,24,25,31,38,39,44,52}},
{60,{1,1,2,4,7,11,16,16,21,27,34,42,51,61,68,74,85,97,110,124}},
{75,{1,2,6,9,17,22,32,43,55,68,84,101,119,138,157,178,206,229,257,284}}}

王守恩 · 发表于 2023-7-8 10:14:58

本帖最后由王守恩于 2023-7-8 10:29 编辑

谢谢 northwolves ！这些按钮我不会用。

Table[2Length@Select[Subsets[Range[2n],{2}],Total@#>n&&#[[1]]^2+n^2≥2Cos[kPi/12]#[[1]]n+#[[2]]^2&&#[[2]]^2+n^2<#[[1]]^2+2n^2#[[2]]&]+Floor[(n Sec[k*Pi/12]/2]-Floor[n/2],{k,1,5},{n,20}]
Table[2Length@Select[Subsets[Range[5n],{2}],Total@#>n&&#[[1]]^2+n^2+2Sin[k*Pi/12]#[[1]]n≥#[[2]]^2&&(#[[2]]^2-#[[1]]^2)^2≥n^2(2Total@#^2-5n^2)&]+Floor[Sqrt@5n/2]-Floor[n/2],{k,0,5},{n,20}]

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  1—5题。上面的公式。
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0},
{0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 5, 7, 7, 9, 11, 13, 15, 16, 19, 20, 23, 26, 30, 32, 32, 34, 38},
{0, 0, 1, 0, 1, 3, 5, 5, 8, 12, 12, 16, 21, 24, 25, 31, 38, 39, 44, 52, 52, 58, 67, 72, 79, 87, 98, 101, 110, 118},
{1, 1, 2, 4, 7, 11, 16, 20, 21, 27, 34, 42, 51, 61, 72, 78, 85, 97, 110, 124, 139, 151, 160, 176, 189, 207, 226, 242, 263, 283},
{1, 2, 6, 9, 17, 22, 32, 43, 55, 68, 84, 101, 119, 138, 157, 178, 206, 229, 257, 284, 316, 351, 381, 414, 452, 487, 531, 568, 612, 650}}
  6—11题。下面的公式。
{1, 3,  8, 14, 21, 29, 40, 52, 66, 82,  99, 117, 138, 160, 185, 209, 239, 265, 294, 326},
{1, 5, 10, 16, 25, 37, 50, 64, 84, 102, 123, 149, 174, 202, 231, 263, 295, 331, 368, 406},
{1, 5, 12, 20, 31, 43, 62, 78, 98, 124, 147, 175, 206, 244, 275, 315, 355, 395, 442, 488},
{1, 5, 12, 22, 37, 53, 70, 88, 114, 142, 173, 205, 240, 280, 319, 369, 417, 463, 514, 568},
{1, 5, 12, 22, 37, 55, 80, 104, 136, 170, 207, 239, 276, 322, 365, 419, 473, 533, 594, 660},
{1, 5, 12, 22, 37, 55, 80, 104, 136, 170, 207, 253, 300, 354, 407, 471, 535, 603, 674, 748}

northwolves · 发表于 2023-7-8 10:24:50

1—5题。有问题。看16, 16,

你加上等号应该就一样了。你的数据包含边上的点

王守恩 · 发表于 2023-7-8 18:41:53

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?
得到一串数(这么简单的一串数，OEIS好像没有，我怀疑是不是OEIS被封了?)。
{1, 1, 2, 4, 7, 11, 16, 20, 21, 27, 34, 42, 51, 61, 72, 78, 85, 97, 110, 124, 139, 151, 160, 176, 189, 207, 226, 242, 263, 283, 296, 318, 337, 361,
386, 408, 435, 453, 476, 504, 525, 555, 578, 610, 639, 663, 692, 726, 753, 789, 822, 860, 887, 921, 956, 992, 1029, 1071, 1110, 1144, ......}
a(01)=01:{1,1},
a(02)=01:{2,1},
a(03)=02:{2,2},{3,3},
a(04)=04:{2,3},{3,2},{3,3},{4,4},
a(05)=07:{2,4},{3,3},{3,4},{4,2},{4,3},{4,4},{5,5},
a(06)=11:{2,5},{3,4},{3,5},{4,3},{4,4},{4,5},{5,2},{5,3},{5,4},{5,5},{6,6},
a(07)=16:{2,6},{3,5},{3,6},{4,4},{4,5},{4,6},{5,3},{5,4},{5,5},{5,6},{6,2},{6,3},{6,4},{6,5},{6,6},{7,7},
a(08)=20:{2,7},{3,6},{3,7},{4,5},{4,6},{5,4},{5,5},{5,6},{5,7},{6,3},{6,4},{6,5},{6,6},{6,7},{7,2},{7,3},{7,5},{7,6},{7,7},{8,8},
a(09)=21:{2,8},{3,7},{4,6},{4,7},{5,5},{5,6},{5,7},{6,4},{6,5},{6,6},{6,7},{7,3},{7,4},{7,5},{7,6},{7,7},{7,8},{8,2},{8,7},{8,8},{9,9},
a(10)=27:{2,9},{3,8},{4,7},{4,8},{5,6},{5,7},{5,8},{6,5},{6,6},{6,7},{6,8},{7,4},{7,5},{7,6},{7,7},{7,8},{8,3},{8,4},{8,5},{8,6},{8,7},{8,8},{8,9},{9,2},{9,8},{9,9},{10,10},

王守恩 · 发表于 2023-7-9 10:20:00

能老老实实的跟着你学一点，就很满足了。谢谢 northwolves ！这些按钮我不会用。

Table[2 Length@Select[Subsets[Range[n], {2}], Total@# > n && #[[1]]^2 + n^2 >= #[[1]] n + #[[2]]^2 < #[[1]]^2 + 2 n^2 #[[2]] &] + Ceiling[n/2], {n, 58}]

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{1, 1, 2, 4, 7, 11, 16, 20, 21, 27, 34, 42, 51, 61, 72, 78, 85, 97, 110, 124, 139, 151, 160, 176, 189, 207, 226, 242, 263, 283, 296, 318, 337, 361,
386, 408, 435, 453, 476, 504, 525, 555, 578, 610, 639, 663, 692, 726, 753, 789, 822, 860, 887, 921, 956, 992, 1029, 1071, 1110, 1144, ......}

northwolves · 发表于 2023-7-9 11:03:27

Table[2Length@Select[Subsets[Range@n,{2}],Total@#>n&&n-#[[1]]>(#[[2]]^2-#[[1]]^2)/n>#[[2]]-n&]+Ceiling[n/2],{n,58}]

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王守恩 · 发表于 2023-7-9 12:36:36

Table[2 Length@Select[Subsets[Range[n],{2}],Total@# > n && #[[1]]^2 - #[[2]]^2≥#[[1]] n-n^2 &]+Ceiling[n/2], {n, 104}]

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[原创] 数字串的通项公式

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