周长相同的梯形有多少个？

王守恩

已知周长=n=1,2,3,4,5,6,7,8,9,...  4条边=上底,左腰,右腰,下底=a,b,c,d=正整数。
(1)。a,b,c,d没有限制, 这样的梯形有多少个？
(2)。a,b,c,d互不相同, 这样的梯形有多少个？
(3)。两条边相同, 这样的梯形有多少个？
(4)。3条边相同, 这样的梯形有多少个？
(5)。直角梯形, 这样的梯形有多少个？
(6)。等腰梯形, 这样的梯形有多少个？
(7)。b<c, 这样的梯形有多少个？
(8)。b≤c, 这样的梯形有多少个？
(9)。...,  这样的梯形有多少个？

譬如:  (8)。b≤c, 这样的梯形有多少个？
a(1)=0,
a(2)=0,
a(3)=0,
a(4)=0,
a(5)=1=1,1,1,2,
a(6)=0,
a(7)=3=1,1,2,3=1,2,2,2=2,1,1,3,
a(8)=1=1,2,2,3,
a(9)=7=1,1,3,4=1,2,2,4=1,2,3,3=1,3,3,2=2,1,2,4=2,2,2,3=3,1,1,4,
a(10)=3=1,2,3,4=1,3,3,3=2,2,2,4,
a(11)=13=1,1,4,5=1,2,3,5=1,2,4,4=1,3,3,4=1,3,4,3=1,4,4,2=2,1,3,5=2,2,2,5=2,2,3,4=2,3,3,3=3,1,2,5=3,2,2,4=4,1,1,5,
a(12)=7=1,2,4,5=1,3,3,5=1,3,4,4=1,4,4,3=2,2,3,5=2,3,3,4=3,2,2,5,
a(13)=22=1,1,5,6=1,2,4,6=1,2,5,5=1,3,3,6=1,3,4,5=1,3,5,4=1,4,4,4=1,4,5,3=1,5,5,2=2,1,4,6=2,2,3,6=2,2,4,5=2,3,3,5=2,3,4,4=2,4,4,3=3,1,3,6=3,2,2,6=3,2,3,5=3,3,3,4=4,1,2,6=4,2,2,5=5,1,1,6,
a(14)=13=1,2,5,6=1,3,4,6=1,3,5,5=1,4,4,5=1,4,5,4=1,5,5,3=2,2,4,6=2,3,3,6=2,3,4,5=2,4,4,4=3,2,3,6=3,3,3,5=4,2,2,6,
a(15)=34
a(16)=22
a(17)=50
a(18)=34
a(19)=70

得到一串数。0, 0, 0, 0, 1, 0, 3, 1, 7, 3, 13, 7, 22, 13, 34, 22, 50, 34, 70, 50, 95, 70, 125, 95, 161, 125, 203, 161, 252, 203, 308, 252, 372, 308, 444, ......

northwolves

前100项：
{0,0,0,0,1,0,3,1,7,3,13,7,22,13,34,22,50,34,70,50,95,70,125,95,161,125,203,161,252,203,308,252,372,308,444,372,525,444,615,525,715,615,825,715,946,825,1078,946,1222,1078,1378,1222,1547,1378,1729,1547,1925,1729,2135,1925,2360,2135,2600,2360,2856,2600,3128,2856,3417,3128,3723,3417,4047,3723,4389,4047,4750,4389,5130,4750,5530,5130,5950,5530,6391,5950,6853,6391,7337,6853,7843,7337,8372,7843,8924,8372,9500,8924,10100,9500}

northwolves

northwolves 发表于 2024-12-28 06:03
前100项：
{0,0,0,0,1,0,3,1,7,3,13,7,22,13,34,22,50,34,70,50,95,70,125,95,161,125,203,161,252,203,308 ...

1. 
   
2. h[r_] := Module[{a = r[[1]], b = r[[2]], c = r[[3]], d = r[[4]]},  a < d && d < a + b + c &&  b^2 > ((d - a)/2 - (c^2 - b^2)/(2 (d - a)))^2];
   
3. v = Table[Table[{x, n - x}, {x, n/2}], {n, 100}];
   
4. y[n_] := Module[{c = 0}, For[k = 2, k < n - 1, k++,  z = Flatten[Table[{x[[1]], y[[1]], y[[2]], x[[2]]}, {x, v[[k]]}, {y, v[[n - k]]}], 1]; q = Select[z, h@# == True &]; c += Length@q]; c];
   
5. Table[y[n], {n, 100}]
   

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northwolves

其他情况修改函数h[r_]的相应参数即可

王守恩

northwolves 发表于 2024-12-28 06:03
前100项：
{0,0,0,0,1,0,3,1,7,3,13,7,22,13,34,22,50,34,70,50,95,70,125,95,161,125,203,161,252,203,308 ...

{0, 0, 0, 0, 1, 0, 3, 1, 7, 3, 13, 7, 22, 13, 34, 22, 50, 34, 70, 50, 95, 70, 125, 95, 161, 125, 203, 161, 252, 203, 308, 252, 372, 308, 444, 372, 525, 444, 615, 525, 715, 615, 825, 715, 946, 825, 1078, \

1. Table[Floor[(n (2 n^2 + 13 - 9 n - 9 (n - 3) (-1)^n) + 12)/192 ], {n, 200}]

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