数字串的通项公式

王守恩

{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}

1. Table[Length@Flatten[Table[NSolve[{m/Sin[a]==n/Sin[b]==k/Sin[a+b]≥(mn/k)/Sin[Pi/3],Pi/3≥a>0,Pi/3≥b>0},{a,b}],{m,1,k},{n,1,k}]]/2,{k,1,30}]

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算式没问题，就是慢了一点。

northwolves

更直观更高效的计算：

1. Table[2Total@Table[Length@Select[Range[n-k+1,n],0<#^2-k^2<=n(n-k)&],{k,n}]+Ceiling[n/2],{n,176}]

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{1,1,2,4,7,11,16,20,21,27,34,42,51,61,72,78,85,97,110,124,139,151,160,176,189,207,226,242,263,283,296,318,337,361,386,408,435,453,476,504,525,555,578,610,639,663,692,726,753,789,822,860,887,921,956,992,1029,1071,1110,1144,1179,1219,1256,1298,1341,1385,1430,1466,1507,1553,1596,1648,1697,1751,1794,1836,1887,1935,1988,2046,2093,2153,2190,2242,2295,2353,2412,2476,2537,2585,2638,2696,2759,2823,2888,2958,3017,3067,3126,3190,3255,3325,3392,3464,3525,3581,3650,3724,3799,3875,3944,4024,4081,4155,4230,4310,4387,4465,4548,4618,4681,4761,4842,4924,5003,5095,5172,5252,5329,5419,5498,5586,5671,5765,5844,5930,6017,6105,6194,6288,6379,6473,6556,6644,6733,6835,6934,7030,7131,7223,7312,7414,7509,7617,7722,7828,7923,8021,8116,8216,8313,8427,8542,8650,8739,8843,8956,9062,9169,9289,9410,9510,9611,9725,9840,9956}

northwolves

或者

1. Table[2Total@Table[Length@Select[Range[Max[k,n-k]+1,n],#^2+n*k<=n^2+k^2&],{k,n}]+Ceiling[n/2],{n,176}]

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1. Table[2Total@Table[Floor@Sqrt[n^2+k^2-n*k]-Max[k,n-k],{k,n}]+Ceiling[n/2],{n,176}]

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1. Table[2Sum[Floor@Sqrt[n^2+k^2-n*k]-Max[k,n-k],{k,n}]+Ceiling[n/2],{n,176}]

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1. Table[2Sum[Floor@Sqrt[n^2+k^2-n*k],{k,n}]-n (3n-1)/2,{n,176}]

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王守恩

332:

1. Table[2Total@Table[Length@Select[Range[n-k,n],(n-k)n≥#^2-k^2>0&],{k,n}]-Floor[(n-3)/2],{n,176}]

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王守恩

每道题(11道)中的#[[1]],#[[2]]都可以去掉吗？ 1—4题勉强可以。第5题不行, 第6题也不行, 7—11题更不行了。

1. Table[2Total@Table[Length@Select[Range[n-k+1,n],(n-2Cos[a\[Pi]/12] k) n ≥ #^2-k^2>0 &],{k,n}]+Floor[(n Sec[a\[Pi]/12])/2]-Floor[n/2],{a,1,4},{n,30}]

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{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}

王守恩

正三角形(边长为整数)内(只要不超出边都可以)摆放三角形ABC(三边为整数), 这样的ABC有几个？

a(1)=01:{1,1,1},
a(2)=02:{1,1,1},{2,2,2},
a(3)=05:{1,1,1},{1,2,2},{2,2,2},{2,2,3},{3,3,3},
a(4)=10:{1,1,1},{1,2,2},{1,3,3},{2,2,2},{2,2,3},{2,3,3},{2,3,4},{3,3,3},{3,3,4},{4,4,4},
a(5)=18:{1,1,1},{1,2,2},{1,3,3},{1,4,4},{2,2,2},{2,2,3},{2,3,3},{2,3,4},{2,4,4},{2,4,5},{3,3,3},{3,3,4},{3,3,5},{3,4,4},{3,4,5},{4,4,4},{4,4,5},{5,5,5},

得到一串数。
{1, 2, 5, 10, 18, 29, 44, 60, 80, 106, 139,

王守恩

正三角形(边长为整数)内(只要不超出边都可以)摆放三角形ABC(三边为整数), 这样的ABC有几个？

a(1)=01:{1,1,1},
a(2)=02:{1,1,1},{2,2,2},
a(3)=05:{1,1,1},{1,2,2},{2,2,2},{2,2,3},{3,3,3},
a(4)=10:{1,1,1},{1,2,2},{1,3,3},{2,2,2},{2,2,3},{2,3,3},{2,3,4},{3,3,3},{3,3,4},{4,4,4},
a(5)=18:{1,1,1},{1,2,2},{1,3,3},{1,4,4},{2,2,2},{2,2,3},{2,3,3},{2,3,4},{2,4,4},{2,4,5},{3,3,3},{3,3,4},{3,3,5},{3,4,4},{3,4,5},{4,4,4},{4,4,5},{5,5,5},

得到一串数, (1)+(2)。
  1, 2, 5, 10, 18, 29, 44, 60, 80, 106, 139, 178, 224, 277, 335, 395, 467,549, 643, 746,

说明: 三角形ABC,   A,B,C的对边为a,b,c,   约定a≤b≤c,
A总可以与正三角形某顶点重合，AB总可以与正三角形某边重合，分2种情形统计
(1)   Pi/3≥B>0 :{1, 2, 4, 7, 12, 19, 29, 41, 54, 70, 90, 114, 143, 177, 217, 260, 307, 360, 420, 487}

1. Table[Length@Flatten[Table[NSolve[{a/Sin[A]==b/Sin[B]==c/Sin[A+B],Pi/3≥A>0,Pi/3≥B>0},{A,B}],{a,1,n},{b,a,n},{c,b,n}]]/2,{n,1,20}]

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(2)Pi/2≥B>Pi/3:{0, 0, 1, 3,  6, 10, 15, 19, 26, 36, 49,   64,   81, 100, 118, 135, 160, 189, 223, 259}

1. Table[Length@Flatten[Table[NSolve[{a/Sin[A]==b/Sin[B]==c/Sin[A+B],((n-c)Sqrt[3])/(2Sin[B-Pi/3])≥a,Pi/3≥A>0,Pi/2≥B>Pi/3},{A,B}],{a,1,n},{b,a+1,n},{c,b,n}]]/2,{n,1,20}]

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(1)=(3)前若干项的和
(3)1, 1, 2, 3, 5, 7, 10, 12, 13, 16, 20, 24, 29, 34, 40, 43, 47, 53, 60, 67, 75, 81, 86, 94, 101, 110, 120, 128, 139, 149}
Table[Sum[Length@Select[Range[n-k,n],(n-k)n≥#^2-k^2>0&],{k,0,n}],{n,1,30}]

王守恩

一个正方形(边长=2,4,6,8,..., n+1个点)格点纸(1*1方格),
去掉一个最大的圆(半径=1,2,3,4,...)，还剩几个格点(包括圆周上的格点)？
a(1)=4*2,
a(2)=4*4,
a(3)=4*6,
a(4)=4*9,
a(5)=4*11,

得到这样一串数: {2, 4, 6, 9, 11, 15, 20, 24, 28, 32, 39, 47, 51, 58, 64, 74, 82, 91, 99, 107}

1. Table[n^2+n-Sum[SquaresR[2,k],{k,2,n^2}]/4,{n,1,20}]

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白新岭

经久不衰的话题，不进门，难懂其道。

王守恩

A135358       
       
6, 12, 19, 25, 32, 38, 45, 51, 58, 64, 71, 77, 83, 90, 96, 103, 109, 116, 122, 129, 135, 142, 148, 154,

\(a(n)=\frac{13n-GCD(n,n+2)}{2}\)   这样简单些。