wayne 发表于 2025-5-16 20:49
@王守恩, 因为王守恩对本公式有改进,特公布一下推导过程.
解方程$(x^2 + y^2 + 2 x y - x - 3 y + 2) = 2...
要是能把取整符号放在表达式的最外面，那才叫牛逼
对于第1个通项公式，你告诉人工智能。按斜着排列，然后顺序增长，
最后人工智能也能得到通项公式
t = 15; Table - n + 1, {m, t}, {n, t - m}] // Grid
1 2 4 7 11 16 22 29 37 46 56 67 79 92
3 5 8 12 17 23 30 38 47 57 68 80 93
6 9 13 18 24 31 39 48 58 69 81 94
10 14 19 25 32 40 49 59 70 82 95
15 20 26 33 41 50 60 71 83 96
21 27 34 42 51 61 72 84 97
28 35 43 52 62 73 85 98
36 44 53 63 74 86 99
45 54 64 75 87 100
55 65 76 88 101
66 77 89 102
78 90 103
91 104
105
图中一共有多少个三角形？
https://bbs.emath.ac.cn/forum.php?mod=viewthread&tid=19750
(出处: 数学研发论坛)
https://bbs.emath.ac.cn/forum.php?mod=redirect&goto=findpost&ptid=19750&pid=102489
一个类似第二个链接的，把取整符号放在最外面的通项公式。
在别人给我一个不存在类似的通项公式的严格的证明之前，
我拒绝相信不存在
iseemu2009 发表于 2025-5-15 09:10
下面是生成数列的程序：
嗨！8楼还藏了个这么好的东东！！
a = f := FoldList]; Grid, {b, 9}]]
{"1", "2", "4", "7", "11", "16", "22", "29", "37"},
{"3", "5", "8", "12", "17", "23", "30", "38", ""},
{"6", "9", "13", "18", "24", "31", "39", "", ""},
{"10", "14", "19", "25", "32", "40", "", "", ""},
{"15", "20", "26", "33", "41", "", "", "", ""},
{"21", "27", "34", "42", "", "", "", "", ""},
{"28", "35", "43", "", "", "", "", "", ""},
{"36", "44", "", "", "", "", "", "", ""},
{"45", "", "", "", "", "", "", "", ""}
改一下！我好像没辙了！！
a = f := FoldList]; Grid, {b, 0, 9}]]
{"0", "0", "1", "3", "6", "10", "15", "21", "28", "36"},
{"2", "3", "5", "8", "12", "17", "23", "30", "38", ""},
{"6", "8", "11", "15", "20", "26", "33", "41", "", ""},
{"12", "15", "19", "24", "30", "37", "45", "", "", ""},
{"20", "24", "29", "35", "42", "50", "", "", "", ""},
{"30", "35", "41", "48", "56", "", "", "", "", ""},
{"42", "48", "55", "63", "", "", "", "", "", ""},
{"56", "63", "71", "", "", "", "", "", "", ""},
{"72", "80", "", "", "", "", "", "", "", ""},
{"90", "", "", "", "", "", "", "", "", ""}
补充内容 (2025-5-18 06:35):
下面的图——通项公式——Table[((m + n)^2 + (m - 2)^2 - 5 n)/2, {m, 9}, {n, 40}]
王守恩 发表于 2025-5-15 15:27
第3个数可以这样出来——Table T] + 1, {T, 39}]
{2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 17 ...
接18楼——第3个数可以这样出来——Table T] + 1, {T, 39}]——如何证明:前后两个数的差只能是 1 或 2 ？？？
{2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 17, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 58, 59, 60, 62, 63, 65, 66, 67, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82,
83, 84, 86, 87, 89, 90, 92, 93, 94, 96, 97, 99, 100, 101, 103, 104, 106, 107, 108, 110, 111, 113, 114, 116, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 133, 134, 135, 137, 138, 140, 141, 142, 144, 145,
147, 148, 149, 151, 152, 154, 155, 157, 158, 159, 161, 162, 164, 165, 166, 168, 169, 171, 172, 174, 175, 176, 178, 179, 181, 182, 183, 185, 186, 188, 189, 191, 192, 193, 195, 196, 198, 199, 200, 202, 203, 205,
206, 207, 209, 210, 212, 213, 215, 216, 217, 219, 220, 222, 223, 224, 226, 227, 229, 230, 232, 233, 234, 236, 237, 239, 240, 240, 241, 243, 244, 246, 247, 248, 250, 251, 253, 254, 256, 257, 258, 260, 261, 263,}
iseemu2009 发表于 2025-5-16 15:37
求新数列的通项公式
“之”字型数列的生成程序如下：
Clear["`*"]
k = 8;(*生成k行完整的“之”字型数列*)
a = Table, Range[(i^2 - i + 2)/2, (i^2 + i)/2], Range[(i^2 + i)/2, (i^2 - i + 2)/2, -1]], {i, k}];
b = Table[], {i, k}];
Grid
本帖最后由 iseemu2009 于 2025-5-19 11:26 编辑
下面程序是47#程序，“之”字型数列的逆算。任意指定数字U，求它在几行几列，增加了视觉效果。第2行的U值可以任意改动，其他字符不要动。
Clear["`*"]
U = 60; (*输入你想要查看的 U 值*)
k = Ceiling[(Sqrt - 1)/2];
{m, n} = If, {1 + (U - (k^2 - k + 2)/2) , k - (U - (k^2 - k + 2)/2) }, {k - (U - (k^2 - k + 2)/2) , 1 + (U - (k^2 - k + 2)/2)}];
a = Table, Range[(i^2 - i + 2)/2, (i^2 + i)/2], Range[(i^2 + i)/2, (i^2 - i + 2)/2, -1]], {i, k}];
b = Table[], {i, k}];
c = ToString[#] <> "行" & /@ Range;
d = Join[{#} & /@ c, # & /@ b, 2];
e = Join[{"表格"}, ToString[#] <> "列" & /@ Range];
f = Join[{e}, d];
Grid}, Spacings -> {1, 1},
Background -> {Automatic, Automatic, {{{m + 1, m + 1}, {1, n}} -> Lighter, {{1, m}, {n + 1, n + 1}} ->
Lighter, {m + 1, n + 1} ->
Lighter}}] // Text
nyy 发表于 2025-5-16 10:37
我来根据你的通项公式，然后由T得到(m,n)的通项公式
用FractionalPart来化简这个，结果失败，不知道有没有人有更好的！
一道相近的趣题——A320040
=========================================================================
n\d| 1 2 3 4 5 6 7 8 9 10 11 12 13
---|---------------------------------------------------------------------
1|1/11/21/31/41/51/61/71/81/91/101/111/121/13
2|2/12/22/32/42/52/62/72/82/92/102/112/122/13
3|3/13/23/33/43/53/63/73/83/93/103/113/123/13
4|4/14/24/34/44/54/64/74/84/94/104/114/124/13
5|5/15/25/35/45/55/65/75/85/95/105/115/125/13
6|6/16/26/36/46/56/66/76/86/96/106/116/126/13
7|7/17/27/37/47/57/67/77/87/97/107/117/127/13
8|8/18/28/38/48/58/68/78/88/98/108/118/128/13
9|9/19/29/39/49/59/69/79/89/99/109/119/129/13
10 | 10/1 10/2 10/3 10/4 10/5 10/6 10/7 10/8 10/9 10/10 10/11 10/12 10/13
11 | 11/1 11/2 11/3 11/4 11/5 11/6 11/7 11/8 11/9 11/10 11/11 11/12 11/13
12 | 12/1 12/2 12/3 12/4 12/5 12/6 12/7 12/8 12/9 12/10 12/11 12/12 12/13
13 | 13/1 13/2 13/3 13/4 13/5 13/6 13/7 13/8 13/9 13/10 13/11 13/12 13/13