整数边长凸等角六边形

王守恩 · 发表于 2025-3-16 08:32:37

我们把网撒得大一点——如果 iseemu2009 不喜欢我就删了——OEIS可没有这样系列资料的。

周长为n, 有a(n)个3边为整数的3边形。
Table[Length@Select[IntegerPartitions[n, {3}], Max[#]2<n &], {n, 3, 100}]
{1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4,7, 5, 8, 7,10, 8, 12, 10, 14, 12, 16,14, 19, 16,21, 19, 24, 21, 27,

周长为n, 有a(n)个4边为整数的4边形。
Table[Length@Select[IntegerPartitions[n, {4}], Max[#]2<n &], {n, 4, 100}]
{1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 12, 16, 18, 23, 24, 31, 33, 41, 43, 53, 55, 67, 69, 83, 86, 102, 104,

周长为n, 有a(n)个5边为整数的5边形。
Table[Length@Select[IntegerPartitions[n, {5}], Max[#]2<n &], {n, 5, 100}]
{1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 23, 25, 35, 39, 52, 57, 74, 81, 103, 111, 139, 150, 184, 197, 239,

周长为n, 有a(n)个6边为整数的6边形。
Table[Length@Select[IntegerPartitions[n, {6}], Max[#]2<n &], {n, 6, 100}]
{1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 28, 37, 46,  59, 71,  91, 107, 134, 157, 193, 222, 271, 308, 371,

周长为n, 有a(n)个7边为整数的7边形。
Table[Length@Select[IntegerPartitions[n, {7}], Max[#]2<n &], {n, 7, 100}]
{1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 70, 93, 112, 145, 171, 218, 256, 320, 372, 458, 528,

周长为n, 有a(n)个8边为整数的8边形。
Table[Length@Select[IntegerPartitions[n, {8}], Max[#]2<n &], {n, 8, 100}]
{1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 104, 134, 167, 211, 258, 322, 389, 480, 572, 698,

周长为n, 有a(n)个9边为整数的9边形。
Table[Length@Select[IntegerPartitions[n, {9}], Max[#]2<n &], {n, 9, 100}]
{1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 50, 69, 87, 116, 145, 189, 233, 299, 363, 458, 553, 687, 820,
......

特别地——把周长为n, 有a(n)个6边为整数的6边形——提出来——与主帖还是有联系的。

Table[Length@Select[IntegerPartitions[n, {6}], Max[#]2<n &], {n, 6, 100}]

{1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 28, 37, 46,  59, 71,  91, 107, 134, 157, 193, 222, 271, 308,  371, 419, 499, 559, 661, 734, 860, 952, 1106, 1216, 1405, 1537, 1764, 1923, 2193, 2381, 2703, 2923,
3301, 3561, 4002, 4302, 4817, 5164, 5758, 6159, 6841, 7300, 8083, 8604, 9495, 10090, 11100, 11771, 12915, 13671, 14958, 15809, 17252, 18204, 19821, 20881, 22683, 23867,  25869, 27180,
29403, 30854, 33311, 34915, 37624, 39390, 42375, 44313, 47589, 49719, 53307, 55635, 59562, 62104, 66388, 69161, 73826, 76843, 81919, 85193, 90701, 94257, 100224, 104070, 110530, ...}

LinearRecurrence[{0, 1, 1, 1, -1, 0, -1, 0, 0, -1, 0, -1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 0, -1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 28, 37, 46}, 100]

不能再短了——33阶——A069907——不能有短一点的了？

王守恩 · 发表于 2025-3-17 08:53:00

这题目挺不错的！！！谢谢 iseemu2009！！！

整数边长凸等角六边形。
{0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 10, 12, 17, 21, 28, 33, 44, 51, 64, 75, 92, 105, 128, 145, 172, 195, 228, 255, 297, 330, 378, 420, 477, 525, 594, 651, 729, 798, 888, 966, 1072, 1162, 1280, 1386, 1520, ...——H(n)
后项-前项=这样一串数。
{0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 02, 05, 04, 07, 05, 11, 07, 13, 11, 17, 13, 023, 017, 027, 023, 033, 027, 042, 033, 048, 042, 057, 048, 069, 057, 078, 069, 090, 078, 106, 0090, 0118, 0106, 0134, ... ——A(n)
再:隔一提一。
{0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658, 707, 763, 812, ... ——B(n)

整数边长凸等角6边形——B(n)。
Table[Total[IntegerPartitions[n, {3}][[;; , 3]]], {1}, {n, 40}]
{0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658},

整数边长凸等角10边形——B(n)。
Table[Total[IntegerPartitions[n, {5}][[;; , 5]]], {2}, {n, 40}]
{0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 48, 62, 78, 98, 122, 149, 181, 219, 262, 314, 370, 436, 510, 595, 691, 797, 916, 1050,1198, 1365, 1545, 1747, 1968, 2212},

整数边长凸等角14边形——B(n)。
Table[Total[IntegerPartitions[n, {7}][[;; , 7]]], {3}, {n, 40}]
{0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 30, 41, 54, 72, 93, 121, 153, 194, 242, 302, 372, 457, 557, 675, 812, 975, 1162, 1381, 1632, 1924, 2254, 2636, 3068, 3562},

整数边长凸等角18边形——B(n)。
Table[Total[IntegerPartitions[n, {9}][[;; , 9]]], {4}, {n, 40}]
{0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 42, 56, 76, 99, 130, 168, 216, 274, 349, 435, 544, 674, 831, 1017, 1244, 1507, 1823, 2194, 2629, 3136, 3734, 4420},

......

这些数字串与11楼还是有联系的。

mathe · 发表于 2025-3-17 10:46:33

这个题目里面序列K(n)代表边长为n的等角六边形数目，其值为
0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 4, 7, 5, 11, 7, 13, 11, 17, 13, 23, 17, 27, 23, 33, 27, 42, 33, 48, 42, 57, 48, 69, 57, 78, 69, 90, 78, 106, 90, 118, 106, 134, 118, 154, 134, 170, 154, 190, 170, 215, 190, 235, 215, 260, 235, 290, 260, 315, 290, 345, 315, 381, 345, 411, 381, 447, 411, 489, 447
特征多项式 x^15 - x^13 - x^12 - x^11 + x^10 + x^8 + x^7 + x^5 - x^4 - x^3 - x^2 + 1
对应A029136

而U(n)对应A001399

王守恩 · 发表于 2025-3-17 12:12:46

接12楼。

整数边长凸等角8边形——B(n)。
Table[Sum[Length@IntegerPartitions[n - 2k, {4}], {k, n/2}], {n, 65}]
{0, 0, 0, 1, 1, 3, 4, 8, 10, 17, 21, 32, 39, 55, 66, 89, 105, 136, 159, 200, 231, 284, 325, 392, 445, 528, 595, 697, 780, 903, 1005, 1152, 1275, 1449, 1596, 1800, 1974, 2211, 2415},

整数边长凸等角12边形——B(n)。
Table[Sum[Length@IntegerPartitions[n - 3k, {6}], {k, n/3}], {n, 65}]
{0, 0, 0, 0, 0, 1, 1, 2, 4, 6, 9, 15, 20, 29, 41, 55, 73, 99, 126, 163, 209, 262, 326, 408, 497, 608, 739, 888, 1062, 1271, 1500, 1771, 2082, 2431, 2828, 3288, 3791, 4368, 5017, 5736},

整数边长凸等角16边形——B(n)。
Table[Sum[Length@IntegerPartitions[n - 4k, {8}], {k, n/4}], {n, 65}]
{0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 37, 53, 70, 98, 126, 169, 216, 284, 356, 457, 568, 718, 881, 1095, 1332, 1637, 1971, 2392, 2859, 3438, 4075, 4854, 5716, 6757, 7903},

整数边长凸等角20边形——B(n)。
Table[Sum[Length@IntegerPartitions[n - 5k, {10}], {k, n/5}], {n, 65}]
{0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, 67, 92, 122, 163, 214, 279, 359, 462, 586, 744, 932, 1166, 1446, 1790, 2199, 2693, 3278, 3980, 4805, 5789, 6935, 8291, 9868},

......

这些数字串与11楼还是有联系的。

王守恩 · 发表于 2025-3-19 07:10:08

小心翼翼的问: 整数边长凸等角五,七,九,...边形有解吗？

iseemu2009 · 发表于 2025-3-19 16:36:57

王守恩发表于 2025-3-19 07:10
小心翼翼的问: 整数边长凸等角五,七,九,...边形有解吗？

对于 n 为奇数的情况，除了整数边长的正 n 边形能满足题目条件外，没有其它凸等角 n 边形存在。
对于 n 为偶数的情况，因为内角相等，可以推出凸等角 n 边形的对边必然平行，那么在正 n 边形的基础上，同时把对边的长度延长相同的整数，新的凸 n 边形同样满足条件。

王守恩 · 发表于 2025-3-21 08:19:54

iseemu2009 发表于 2025-3-19 16:36
对于 n 为奇数的情况，除了整数边长的正 n 边形能满足题目条件外，没有其它凸等角 n 边形存在。
对于 n ...

也就是说: a,b = 正整数, x,y 无正整数解。
Table[Solve[{(x*Sin[(2 a + 3) Pi] + 2 y)/Cos[(2 a + 3) Pi] == b, b > x > 0, y > 0}, {x, y}, Integers], {a, 30}, {b, 4 a}]

王守恩 · 发表于 6 天前

等角不等边(整数)凸六边形——这个"凸"字可以去掉吗？

k(21)=2: {1,2,3,4,5,6},{1,3,5,2,4,6},
k(22)=1: {1,2,5,3,4,7},
k(23)=1: {1,3,6,2,4,7},
k(24)=2: {1,2,3,5,6,7},{1,2,6,3,4,8},
k(25)=2: {1,3,7,2,4,8},{1,4,6,2,5,7},
k(26)=3: {1,2,4,5,6,8},{1,2,7,3,4,9},{1,4,5,3,6,7},
k(27)=7: {1,2,3,6,7,8},{1,2,6,4,5,9},{1,3,5,4,6,8},{1,3,8,2,4,9},{1,4,7,2,5,8},{2,3,4,5,6,7},{2,4,6,3,5,7},
k(28)=3: {1,2,8,3,4,10},{1,3,4,5,7,8},{2,3,6,4,5,8},
k(29)=6: {1,2,4,6,7,9},{1,2,7,4,5,10},{1,3,9,2,4,10},{1,4,8,2,5,9},{1,5,7,2,6,8},{2,4,7,3,5,8},
k(30)=6: {1,2,3,7,8,9},{1,2,9,3,4,11},{1,4,7,3,6,9},{1,5,6,3,7,8},{2,3,4,6,7,8},{2,3,7,4,5,9},
k(31)=9: {1,2,5,6,7,10},{1,2,8,4,5,11},{1,3,4,6,8,9},{1,3,7,4,6,10},{1,3,10,2,4,11},{1,4,9,2,5,10},{1,5,8,2,6,9,},{2,4,8,3,5,9},{2,5,7,3,6,8},
k(32)=8: {1,2,4,7,8,10},{1,2,7,5,6,11},{1,2,10,3,4,12},{1,3,6,5,7,10},{1,4,8,3,6,10},{2,3,5,6,7,9},{2,3,8,4,5,10},{2,5,6,4,7,8},

得到这样一串数——2, 1, 1, 2, 2, 3, 7, 3, 6, 6, 9, 8, 16, 11, 15, 15, 21, 18, 30, 22, 33, 31, 39, 34, 51, 42, 57, 53, 66, 58, 83, 69, 90, 84,——这可是OEIS没有的。

王守恩 · 发表于 6 小时前

这题目挺不错的——谢谢 iseemu2009——想你——再给我们出题目！
等角不等边(边长只能用1,2,3,4,5,6,7,8,9九个数)六边形有53个。谢谢 “浩瀚宇宙” ！
01: (1,4,5,2,3,6)——依次连接,首尾连接——等角不等边六边形
02: (1,4,5,3,2,7)
03: (1,4,6,2,3,7)
04: (1,4,6,3,2,8)
05: (1,4,7,2,3,8)
06: (1,4,7,3,2,9)
07: (1,4,8,2,3,9)
08: (1,5,3,4,2,6)
09: (1,5,6,2,4,7)
10: (1,5,6,4,2,9)
11: (1,5,7,2,4,8)
12: (1,5,8,2,4,9)
13: (1,6,3,5,2,7)
14: (1,6,4,5,2,8)
15: (1,6,5,3,4,7)
16: (1,6,5,4,3,8)
17: (1,6,7,2,5,8)
18: (1,6,7,3,4,9)
19: (1,6,8,2,5,9)
10: (1,7,3,6,2,8)
21: (1,7,4,5,3,8)
22: (1,7,4,6,2,9)
23: (1,7,6,3,5,8)
24: (1,7,8,2,6,9)
25: (1,8,3,7,2,9)
26: (1,8,4,6,3,9)
27: (1,8,6,4,5,9)
28: (1,8,7,3,6,9)
29: (2,5,6,3,4,7)
30: (2,5,6,4,3,8)
31: (2,5,7,3,4,8)
32: (2,5,7,4,3,9)
33: (2,5,8,3,4,9)
34: (2,6,4,5,3,7)
35: (2,6,7,3,5,8)
36: (2,6,8,3,5,9)
37: (2,7,4,6,3,8)
38: (2,7,5,6,3,9)
39: (2,7,6,4,5,8)
40: (2,7,6,5,4,9)
41: (2,7,8,3,6,9)
42: (2,8,4,7,3,9)
43: (2,8,5,6,4,9)
44: (2,8,7,4,6,9)
45: (3,6,7,4,5,8)
46: (3,6,7,5,4,9)
47: (3,6,8,4,5,9)
48: (3,7,5,6,4,8)
49: (3,7,8,4,6,9)
50: (3,8,5,7,4,9)
51: (3,8,7,5,6,9)
52: (4,7,8,5,6,9)
53: (4,8,6,7,5,9)

iseemu2009 · 发表于 3 小时前

王守恩发表于 2025-3-31 09:03
这题目挺不错的——谢谢 iseemu2009——想你——再给我们出题目！
等角不等边(边长只能用1,2,3,4,5,6,7,8,9 ...

你都快成有趣题目的开发者了

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[原创] 整数边长凸等角六边形