数字串的通项公式

王守恩

  1, 5, 10, 14, 19, 23, 28, 32, 37, 41, 46, 50, 55, 59, 64, 68, 73,77, 82, 86, 91, 95, 100, 104, 109, 113,
  1, 0, 5, 0, 10, 0, 14, 0, 19, 0, 23, 0, 28, 0, 32, 0, 37, 0, 41, 0, 46, 0, 50, 0, 55, 0, 59, 0, 64, 0, 68, 0, 73,
  1, 0, 0, 5, 0, 0, 10, 0, 0, 14, 0, 0, 19, 0, 0, 23, 0, 0, 28, 0, 0, 32, 0, 0, 37, 0, 0, 41, 0,  0, 46, 0, 0, 50, 0, 0,                 
  1, 0, 0, 0, 5, 0, 0, 0, 10, 0, 0, 0, 14, 0, 0, 0, 19, 0, 0, 0, 23, 0, 0, 0, 28, 0, 0, 0, 32, 0, 0, 0, 37, 0, 0, 0, 41, 0,
   

northwolves

王守恩 发表于 2023-7-3 19:34
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？

得到一串数: 1,3,8, ...

a(8)=50
{{1,8},{2,7},{2,8},{3,6},{3,7},{3,8},{4,5},{4,6},{4,7},{4,8},{5,4},{5,5},{5,6},{5,7},{5,8},{5,9},{6,3},{6,4},{6,5},{6,6},{6,7},{6,8},{6,9},{7,2},{7,3},{7,4},{7,5},{7,6},{7,7},{7,8},{7,9},{7,10},{8,1},{8,2},{8,3},{8,4},{8,5},{8,6},{8,7},{8,8},{8,9},{8,10},{8,11},{9,5},{9,6},{9,7},{9,8},{10,7},{10,8},{11,8}}

northwolves

1. Table[Length@Select[Tuples[Range@(Sqrt[2]*n),{2}],Total@#>n&&#[[1]]^2-n^2<#[[2]]^2<(#[[1]]^2+n^2)&&(#[[1]]^2-#[[2]]^2)^2+5n^4>2(#[[1]]^2+#[[2]]^2)*n^2&],{n,40}]

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{1,3,8,12,21,29,40,50,66,82,99,113,138,160,183,207,239,265,294,324,361,399,434,466,511,558,597,639,692,740,789,837,894,953,1004,1062,1125,1185,1254,1314,1385,1447,1523,1591,1664,1742,1821,1897,1976,2058}

northwolves

王守恩 发表于 2023-7-4 14:14
1, 5, 10, 14, 19, 23, 28, 32, 37, 41, 46, 50, 55, 59, 64, 68, 73,77, 82, 86, 91, 95, 100, 104, 109 ...

1. Array[CoefficientList[Series[(2x^#+1)^2/((x^#-1)^2(x^#+1)),{x,0,29}],x]&,4]

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$\{\{1,5,10,14,19,23,28,32,37,41,46,50,55,59,64,68,73,77,82,86,91,95,100,104,109,113,118,122,127,131\},\{1,0,5,0,10,0,14,0,19,0,23,0,28,0,32,0,37,0,41,0,46,0,50,0,55,0,59,0,64\},\{1,0,0,5,0,0,10,0,0,14,0,0,19,0,0,23,0,0,28,0,0,32,0,0,37,0,0,41\},\{1,0,0,0,5,0,0,0,10,0,0,0,14,0,0,0,19,0,0,0,23,0,0,0,28,0,0,0,32\}\}$

王守恩

northwolves 发表于 2023-7-4 14:57
$\{\{1,5,10,14,19,23,28,32,37,41,46,50,55,59,64,68,73,77,82,86,91,95,100,104,109,113,118,122 ...

1. Table[CoefficientList[Series[(2 x^n + 1)^2/((x^n + 1) (x^n - 1)^2), {x, 0, 29 n}], x], {n,1, 4}]

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{{1, 5, 10, 14, 19, 23, 28, 32, 37, 41, 46, 50, 55, 59, 64, 68, 73, 77, 82, 86, 91, 95, 100, 104, 109, 113, 118, 122, 127, 131},
{1, 0, 5, 0, 10, 0, 14, 0, 19, 0, 23, 0, 28, 0, 32, 0, 37, 0, 41, 0, 46, 0, 50, 0, 55, 0, 59, 0, 64, 0, 68, 0, 73, 0, 77, 0, 82, 0, 86, 0, 91, 0, 95, 0, 100, 0, 104, 0, 109, 0, 113, 0, 118, 0, 122, 0, 127, 0, 131},
{1, 0, 0, 5, 0, 0, 10, 0, 0, 14, 0, 0, 19, 0, 0, 23, 0, 0, 28, 0, 0, 32, 0, 0, 37, 0, 0, 41, 0, 0, 46, 0, 0, 50, 0, 0, 55, 0, 0, 59, 0, 0, 64, 0, 0,
68, 0, 0, 73, 0, 0, 77, 0, 0, 82, 0, 0, 86, 0, 0, 91, 0, 0, 95, 0, 0, 100, 0, 0, 104, 0, 0, 109, 0, 0, 113, 0, 0, 118, 0, 0, 122, 0, 0, 127, 0, 0, 131},
{1, 0, 0, 0, 5, 0, 0, 0, 10, 0, 0, 0, 14, 0, 0, 0, 19, 0, 0, 0, 23, 0, 0, 0, 28, 0, 0, 0, 32, 0, 0, 0, 37, 0, 0, 0, 41, 0, 0, 0, 46, 0, 0, 0, 50, 0, 0, 0, 55, 0, 0, 0, 59, 0, 0, 0, 64, 0, 0, 0,
68, 0, 0, 0, 73, 0, 0, 0, 77, 0, 0, 0, 82, 0, 0, 0, 86, 0, 0, 0, 91, 0, 0, 0, 95, 0, 0, 0, 100, 0, 0, 0, 104, 0, 0, 0, 109, 0, 0, 0, 113, 0, 0, 0, 118, 0, 0, 0, 122, 0, 0, 0, 127, 0, 0, 0, 131}}

northwolves

1. Table[Length@Select[Tuples[Range@(Sqrt[2]*n),{2}],Total@#>n&&#[[1]]^2-n^2<=#[[2]]^2<=(#[[1]]^2+n^2)&&(#[[1]]^2-#[[2]]^2)^2+5n^4>2(#[[1]]^2+#[[2]]^2)*n^2&],{n,100}]

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{1,3,8,14,21,29,40,52,66,82,99,117,138,160,185,209,239,265,294,326,363,399,434,472,511,558,597,641,692,742,789,839,894,953,1006,1066,1125,1185,1254,1318,1385,1449,1523,1593,1668,1742,1821,1903,1976,2058,2146,2234,2313,2399,2502,2592,2677,2771,2872,2973,3064,3170,3279,3383,3488,3592,3705,3820,3931,4045,4166,4282,4399,4521,4644,4768,4900,5028,5155,5285,5426,5560,5685,5829,5975,6115,6254,6396,6555,6693,6842,6994,7155,7306,7455,7619,7784,7946,8103,8265}

northwolves

整数的运算比小数还是要快得多

northwolves

用 Subsets还能再稍快些：

1. Table[Floor[Sqrt[5]*n/2]-Ceiling[(n-1)/2]+2Length@Select[Select[Subsets[Range@(Sqrt[2]*n),{2}],Total@#>n&&#[[2]]^2<=#[[1]]^2+n^2&],(#[[1]]^2-#[[2]]^2)^2+5n^4>2(#[[1]]^2+#[[2]]^2)*n^2&],{n,100}]

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王守恩

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好像没有): 1,1,2,4,7,11,16,20,21,27,34,42,51,61,72,78,85,97,110,124,...,

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

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王守恩

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):1,3,10,16,29,37,48,68,84,110,127,147,182,206,245,273,315,347,380,432,...,

1. 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,Sqrt[2]k},{n,1,Sqrt[2]k}]]/2,{k,1,20}]

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算式没问题，就是太慢了。