将6颗红珠跟6颗黄珠穿成一个好圆环

王守恩

将6颗红珠跟6颗黄珠穿成一个圆环,  若任意颗连续相邻的珠子中,红珠跟黄珠最多相差3颗,  
就称这个圆环为好圆环,  好圆环有多少种？

王守恩

将1颗红珠跟1颗黄珠穿成一个圆环,  好圆环有1种。
01, 01,

将2颗红珠跟2颗黄珠穿成一个圆环,  好圆环有2种。
01, 0011,
02, 0101,

将3颗红珠跟3颗黄珠穿成一个圆环,  好圆环有3种。
01, 000111,
02, 001011,
03, 010101,

将4颗红珠跟4颗黄珠穿成一个圆环,  好圆环有7种。
01, 00010111,
02, 00011011,
03, 00100111
04, 00101011,
05, 00101101,
06, 00110011,
07, 01010101,

将5颗红珠跟5颗黄珠穿成一个圆环,  好圆环有13种。
01, 0001010111,
02, 0001011011,
03, 0001011101,
04, 0001100111,
05, 0001101011,
06, 0010010111,
07, 0010011011,
08, 0010011101,
09, 0010100111,
10, 0010101011,
11, 0010101101,
12, 0010110011,
13, 0101010101,

王守恩

将6颗红珠跟6颗黄珠穿成一个圆环,  好圆环有31种。
01, 000101010111,
02, 000101011011,
03, 000101011101,
04, 000101100111,
05, 000101101011,
06, 000101101101
07, 000101110011
08, 000110010111
09, 000110011011
10, 000110100111
11, 000110101011
12, 000110110011
13, 000111000111
14, 001001010111
15, 001001011011
16, 001001011101
17, 001001100111
18, 001001101011
19, 001010010111,
20, 001010011011,
21, 001010011101,
22, 001010100111,
23, 001010101011,
24, 001010101101,
25, 001010110011,
26, 001010110101
27, 001010111001
28, 001011001011
29, 001011001101
30, 001011010011
31, 010101010101,

补充内容 (2024-6-9 16:47):
漏了一个:  001100110011,

王守恩

将7颗红珠跟7颗黄珠穿成一个圆环,  好圆环有66种。
01, 00010101010111,
02, 00010101011011,
03, 00010101011101,
04, 00010101100111,
05, 00010101101011,
06, 00010101101101,
07, 00010101110011,
08, 00010101110101,
09, 00010110010111,
10, 00010110011011,
11, 00010110011101,
12, 00010110100111,
13, 00010110101011,
14, 00010110101101,
15, 00010110110011,
16, 00010111000111,
17, 00010111001011,
18, 00010111010011,
19, 00011001010111,
20, 00011001011011,
21, 00011001100111,
22, 00011001101011,
23, 00011001110011,
24, 00011010010111,
25, 00011010011011,
26, 00011010100111,
27, 00011010101011,
28, 00011011000111,
29, 00011100010111,
30, 00011100100111,
31, 00100101010111,
32, 00100101011011,
33, 00100101011101,
34, 00100101100111,
35, 00100101101011,
36, 00100101011101,
37, 00100101100111,
38, 00100101101011,
39, 00100101101101,
40, 00100101110011,
41, 00100101110101,
42, 00100110010111,
43, 00100110011011,
44, 00100110011101,
45, 00100110100111,
46, 00100110101011,
47, 00100110110011,
48, 00101001010111,
49, 00101001011011,
50, 00101001011101,
51, 00101001100111,
52, 00101001101011,
53, 00101001101101,
54, 00101001110011,
55, 00101010010111,
56, 00101010011011,
57, 00101010100111,
58, 00101010101011,
59, 00101010101101,
60, 00101010110011,
61, 00101010110101,
62, 00101011001011,
63, 00101011001101,
64, 00101011010011,
65, 00101100110011
66, 01010101010101,

王守恩

这题目真还挺诱人的！

根据前面几项:    a(1)=1, a(2)=2, a(3)=3, a(4)=7, a(5)=13, a(6)=32, a(7)=66+?, ...

OEIS有3个对应的数字串,  可惜都不是什么"善桩"!

A324844——1, 2, 3, 7, 13, 32, 71, 170, 406, 1002, 2469,  6204, 15644, 39871, 102116, 263325,  682079,
A007827——1, 2, 3, 7, 13, 32, 73, 190, 488, 1350, 3741, 10765, 31311, 92949, 278840, 847511, 2599071,
A250308——1, 2, 3, 7, 13, 32, 74, 192, 497, 1379, 3844, 11111, 32500, 96977, 292600, 894353, 2758968,

将7颗红珠跟7颗黄珠穿成一个圆环,  好圆环有72种。
01, 00010101010111,
02, 00010101011011,
03, 00010101011101,
04, 00010101100111,
05, 00010101101011,
06, 00010101101101,
07, 00010101110011,
08, 00010101110101,
09, 00010110010111,
10, 00010110011011,
11, 00010110011101,
12, 00010110100111,
13, 00010110101011,
14, 00010110101101,
15, 00010110110011,
16, 00010111000111,
17, 00010111001011,
18, 00010111010011,
19, 00011001010111,
20, 00011001011011,
21, 00011001100111,
22, 00011001101011,
23, 00011001110011,
24, 00011010010111,
25, 00011010011011,
26, 00011010100111,
27, 00011010101011,
28, 00011011000111,
29, 00011100010111,
30, 00011100100111,
31, 00100101010111,
32, 00100101011011,
33, 00100101011101,
34, 00100101100111,
35, 00100101101011,
36, 00100101011101,
37, 00100101100111,
38, 00100101101011,
39, 00100101101101,
40, 00100101110011,
41, 00100101110101,
42, 00100110010111,
43, 00100110011011,
44, 00100110011101,
45, 00100110100111,
46, 00100110101011,
47, 00100110110011,
48, 00100111000111
49, 00100111001101
50, 00100111010011
51, 00100111010101
52, 00101001010111,
53, 00101001011011,
54, 00101001011101,
55, 00101001100111,
56, 00101001101011,
57, 00101001101101,
58, 00101001110011,
59, 00101001110101,
60, 00101010010111,
61, 00101010011011,
62, 00101010011101
63, 00101010100111,
64, 00101010101011,
65, 00101010101101,
66, 00101010110011,
66, 00101010110101,
67, 00101011001011,
68, 00101011001101,
69, 00101011010011,
70, 00101011010101,
71, 00101100110011
72, 01010101010101,       

王守恩

参考类似的题目。

将n颗2种颜色的珠子穿成一个圆环。
A000029 ——1, 2, 3, 4, 6, 8, 13, 18, 30, 46, 78, 126, 224, 380, 687, 1224, 2250, 4112, 7685, 14310, 27012, 50964, 96909, 184410, 352698, 675188, 1296858, 2493726, 4806078, 9272780}

1. Table[Total[EulerPhi[n/#] 2^# & /@ Divisors[n]/(2 n)] + (2^Ceiling[n/2] + 2^Ceiling[(n + 1)/2])/4, {n, 35}]

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将n颗红珠跟n颗黄珠穿成一个圆环,  若任意颗连续相邻的珠子中,红珠跟黄珠最多相差2颗。
A053656——1, 2, 2, 4, 4, 9, 10, 22, 30, 62, 94, 192, 316, 623, 1096, 2122, 3856, 7429, 13798, 26500, 49940, 95885, 182362, 350650, 671092, 1292762, 2485534, 4797886, 9256396, 17904476}

1. Table[Total[EulerPhi[n/#] 2^# & /@ Divisors[n]/(2 n)] + 2^(n/2 - 2) Mod[n - 1, 2], {n, 35}]

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这个公式与上面的公式有类似,  与A053656的公式也有不同。
A053656——好像没有递推公式, 可以有吗？ 谢谢各位好友！

主帖(红珠跟黄珠最多相差3颗)比上面的题目还是要难一些。