有几种走法？

王守恩

递推有了，通项也可以有吗？谢谢！
正m边形, 走了n条路, 有S(m,n)种走法。数据没问题。

1. Table[RecurrenceTable[{(m-1)(3+n)b[3+n]==(12m-6+(6m-4)n)b[2+n]+m(3m-12+(m-8)n)b[1+n]-m^2(6+4n)b[n],b[1]==0,b[2]==1,b[3]==2},b,{n,1,22}],{m,3,20}]

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S(03,n)=0, 1, 2, 8, 26, 93, 330, 1194, 4352, 15998, 59180, 220138, 822718, 3087325, 11626986, 43926498, 166422096, 632113134, 2406419484, 9180135960, 35087235348, 134339184354,
S(04,n)=0, 1, 2, 9, 30, 114, 420, 1585, 5990, 22806, 87164, 334474, 1287468, 4969476, 19226952, 74544129, 289542198, 1126466286, 4388922252, 17122525134, 66879612228, 261510344316,  
S(05,n)=0, 1, 2, 10, 34, 137, 522, 2054, 8040, 31722, 125356, 496956, 1973862, 7854905, 31305290, 124932670, 499150320, 1996293790, 7991014300, 32012556140, 128334199620, 514797408170,
S(06,n)=0, 1, 2, 11, 38, 162, 636, 2607, 10550, 43118, 176084, 721294, 2957308, 12142276, 49899192, 205243479, 844786998, 3479312886, 14337289572, 59107095114, 243772689108, 1005732285276,  
S(07,n)=0, 1, 2, 12, 42, 189, 762, 3250, 13568, 57390, 241916, 1023358, 4328814, 18334173, 77685978, 329380554, 1397097168, 5928227118, 25162550364, 106831928448, 453674640948, 1926957200034,
S(08,n)=0, 1, 2, 13, 46, 218, 900, 3989, 17142, 74958, 325660, 1421298, 6196428, 27051860, 118103816, 515897893, 2253950406, 9850146694, 43053933484, 188215008710, 822903965988, 3598251106604,  
S(09,n)=0, 1, 2, 14, 50, 249, 1050, 4830, 21320, 96266, 430364, 1935664, 8686678, 39050761, 175482522, 789012774, 3547619568, 15954692766, 71758002012, 322775860884, 1451974942308, 6531990928266,
S(10,n)=0, 1, 2, 15, 54, 282, 1212, 5779, 26150, 121782, 559316, 2589526, 11946012, 55236660, 255184440, 1179739995, 5453098470, 25211813790, 116563164900, 538964157090, 2492115056820, 11523805619820,
S(11,n)=0, 1, 2, 16, 58, 317, 1386, 6842, 31680, 151998, 716044, 3408594, 16142238, 76682621, 363755402, 1727152114, 8197639248, 38920382446, 184768512412, 877250628584, 4164993971028, 19775272715426,  
S(12,n)=0, 1, 2, 17, 62, 354, 1572, 8025, 37958, 187430, 904316, 4421338, 21465964, 104646628, 509085768, 2479770729, 12071590038, 58788681918, 286250881164, 1393984007358, 6788107176708, 33056709366684,  
S(13,n)=0, 1, 2, 18, 66, 393, 1770, 9334, 45032, 228618, 1128140, 5659108, 28132038, 140589945, 700581546, 3497093838, 17440613616, 87027930654, 434142688284, 2166138576060, 10806929431428, 53919426525354,
S(14,n)=0, 1, 2, 19, 70, 434, 1980, 10775, 52950, 276126, 1391764, 7156254, 36380988, 186196196, 949345592, 4851264319, 24759367638, 126460035046, 645633799972, 3297088154234, 16835086845588, 85968249510716,
S(15,n)=0, 1, 2, 20, 74, 477, 2202, 12354, 61760, 330542, 1699676, 8950246, 46480462, 243391165, 1268368890, 6628884570, 34586727120, 180640799790, 942911425500, 4923553961040, 25703937532020, 134206664351970,
S(16,n)=0, 1, 2, 21, 78, 522, 2436, 14077, 71510, 392478, 2056604, 11081794, 58726668, 314363316, 1672731912, 8932982349, 47602629798, 253999866006, 1354252753452, 7223885769654, 38523099026148, 205468630640076,
S(17,n)=0, 1, 2, 22, 82, 569, 2682, 15950, 82248, 462570, 2467516, 13594968, 73445814, 401585033, 2179816058, 11885132854, 64626625008, 351998687518, 1915285772764, 10427848106468, 56753616239268, 308952911789834,
S(18,n)=0, 1, 2, 23, 86, 618, 2940, 17979, 94022, 541478, 2937620, 16537318, 90995548, 507834580, 2809525176, 15627742083, 86638206726, 481307896014, 2670434446884, 14828095949010, 82295626519668, 456877716230604,
S(19,n)=0, 1, 2, 24, 90, 669, 3210, 20170, 106880, 629886, 3472364, 19959994, 111766398, 636218781, 3584517162, 20326496514, 114799011408, 650005446126, 3674565135132, 20793537440184, 117592548466068, 665275521741666,
S(20,n)=0, 1, 2, 25, 94, 722, 3492, 22529, 120870, 728502, 4077436, 23917866, 136183212, 790196420, 4530445640, 26172984145, 150476961270, 867796971790, 4994851881100, 28784794562990, 165753998465220, 954950173554620,

aimisiyou

王守恩 发表于 2024-1-7 19:15
递推有了，通项也可以有吗？谢谢！
正m边形, 走了n条路, 有S(m,n)种走法。数据没问题。

递推式正确吗？

王守恩

aimisiyou 发表于 2024-1-10 19:37
递推式正确吗？

41楼的题目是这样。
m边形, n条路, 有S(m,n)种走法。
环形路不限制,也就是取S(m,n)的最大值。
基本思路：
1环可以确定S(m,2),S(m,3);
2环可以确定S(m,4),S(m,5);
3环可以确定S(m,6),S(m,7);
4环可以确定S(m,8),S(m,9);
......
每环完成5个数: 2个数是任务, 3个数作铺垫。具体：
1环: 确定 S(m,1),  S(m,2),  S(m,3),  S(m,4),  S(m,5);
2环: 前面的数同1环, S(m,4)=1环的S(m,4)+1, S(m,5)=1环的S(m,5)+06, 由这5个数可以推出后面的数;
3环: 前面的数同2环, S(m,6)=2环的S(m,6)+1, S(m,7)=2环的S(m,7)+10, 由这7个数可以推出后面的数;
4环: 前面的数同3环, S(m,8)=3环的S(m,8)+1, S(m,9)=3环的S(m,9)+14, 由这9个数可以推出后面的数;
......
闹着玩的。祝大家新年快乐！生活是不可能压垮我们的。

aimisiyou

王守恩 发表于 2024-1-11 11:31
41楼的题目是这样。
m边形, n条路, 有S(m,n)种走法。
环形路不限制,也就是取S(m,n)的最大值。

邻接矩阵的幂。

王守恩

"1"是最重要的。这里给出 1 环的数据。41#给出的是无限环的数据。
S(3,n)=0, 1, 2, 07, 20, 61, 182,  547, 1640, 4921, 14762, 44287, 132860, 398581, 1195742, 3587227, 10761680, 32285041, 96855122, 290565367, 871696100, 2615088301,  
S(4,n)=0, 1, 2, 08, 24,  80, 256,  832, 2688, 8704, 28160,  91136, 294912, 954368, 3088384, 9994240, 32342016, 104660992, 338690048, 1096024064, 3546808320, 11477712896,
S(5,n)=0, 1, 2, 09, 28, 101, 342, 1189, 4088, 14121, 48682,167969, 579348, 1998541, 6893822, 23780349, 82029808, 282961361, 976071762, 3366950329, 11614259468, 40063270581,
S(6,n)=0, 1, 2, 10, 32, 124, 440, 1624, 5888, 21520, 78368, 285856, 1041920, 3798976, 13849472, 50492800, 184082432, 671121664, 2446737920, 8920205824, 32520839168, 118562913280,
S(7,n)=0, 1, 2, 11, 36, 149, 550, 2143, 8136, 31273, 119498, 457907, 1752300, 6709949, 25685998, 98341639, 376485264, 1441362001, 5518120850, 21125775707, 80878397364, 309637224677,
S(8,n)=0, 1, 2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872, 183251763200, 733008101376,  
S(9,n)=0, 1, 2, 13, 44, 205, 806, 3457, 14168, 59449, 246410, 1027861, 4273412, 17797573, 74055854, 308289865, 1283082416, 5340773617, 22229288978, 92525540509, 385114681820, 1602959228221,

1. Table[ChebyshevU[n,-I/Sqrt[a]]*(Sqrt[a]/-I)^n,{a,3,9},{n,0,20}] // FullSimplify

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