通项公式

王守恩

3条边(a,b,c)都是整数的三角形，我们取面积最大的,
当a+b+c=n,  n=11,12,13,14,15,16,17,18, ......
请您写出用n来表示3条边的通项公式。

王守恩

从1~365中取2个不同数, 2个数之和恰好等于366有182组。(366是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{363}\frac{x^{366}-x^i}{x^{i+182}-x^{182}},(x,0,364)\bigg],x\bigg]\)[[364]]

从1~365中取12个不同数, 12个数之和恰好等于2196有 10,655,873,125,881,071,133组。(2196是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{353}\frac{x^{366}-x^i}{x^{i+177}-x^{177}},(x,0,2119)\bigg],x\bigg]\)[[2119]]

从1~365中取22个不同数, 22个数之和恰好等于4026有 90,400,006,783,955,844,446,363,021,686,652组。(4026是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{343}\frac{x^{366}-x^i}{x^{i+172}-x^{172}},(x,0,3774)\bigg],x\bigg]\)[[3774]]

从1~365中取32个不同数, 32个数之和恰好等于5856有 6,429,619,212,767,264,081,938,214,593,840,653,588,467,393组。(5856是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{333}\frac{x^{366}-x^i}{x^{i+167}-x^{167}},(x,0,5329)\bigg],x\bigg]\)[[5329]]

从1~365中取42个不同数, 42个数之和恰好等于7686有 15,628,610,385,975,443,405,198,520,232,956,659,310,953,492,784,582,040组。(7686是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{323}\frac{x^{366}-x^i}{x^{i+162}-x^{162}},(x,0,6784)\bigg],x\bigg]\)[[6784]]

从1~365中取52个不同数, 52个数之和恰好等于9516有 2,670,630,333,002,279,960,845,281,844,182,378,854,847,648,865,674,133,712,853,542组。(9516是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{313}\frac{x^{366}-x^i}{x^{i+157}-x^{157}},(x,0,8139)\bigg],x\bigg]\)[[8139]]

从1~365中取62个不同数, 62个数之和恰好等于11346有 49,748,750,626,982,865,763,986,416,940,067,779,162,302,652,366,469,638,594,637,743,082,250组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{303}\frac{x^{366}-x^i}{x^{i+152}-x^{152}},(x,0,9394)\bigg],x\bigg]\)[[9394]]

从1~365中取72个不同数, 72个数之和恰好等于13176有 135,481,907,930,175,289,355,266,190,568,641,679,304,610,330,180,572,889,020,549,945,144,295,033,233组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{293}\frac{x^{366}-x^i}{x^{i+147}-x^{147}},(x,0,10549)\bigg],x\bigg]\)[[10549]]

从1~365中取82个不同数, 82个数之和恰好等于15006有 66,450,206,598,429,682,628,568,057,932,527,035,042,767,873,072,888,187,003,576,384,540,964,503,149,956,526组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{283}\frac{x^{366}-x^i}{x^{i+142}-x^{142}},(x,0,11604)\bigg],x\bigg]\)[[11604]]

从1~365中取92个不同数, 92个数之和恰好等于16836有 6,849,865,466,092,346,029,128,136,216,200,617,602,021,866,965,346,024,560,761,533,508,061,452,355,694,620,310,906组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{273}\frac{x^{366}-x^i}{x^{i+137}-x^{137}},(x,0,12559)\bigg],x\bigg]\)[[12559]]

从1~365中取102个不同数,10 2个数之和恰好等于18666有 166,906,812,106,900,780,203,168,662,504,043,192,206,214,971,270,047,654,796,119,609,935,328,361,382,023,100,614,735,540组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{263}\frac{x^{366}-x^i}{x^{i+132}-x^{132}},(x,0,13414)\bigg],x\bigg]\)[[13414]]

从1~365中取112个不同数, 112个数之和恰好等于20496有 1,053,013,743,772,085,687,318,774,904,578,338,951,685,097,138,572,258,364,487,025,174,930,071,037,936,299,865,479,215,841,832组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{253}\frac{x^{366}-x^i}{x^{i+127}-x^{127}},(x,0,14169)\bigg],x\bigg]\)[[14169]]

从1~365中取122个不同数, 122个数之和恰好等于22326有 1,847,357,036,944,212,468,060,799,371,115,931,445,420,684,557,057,887,411,515,479,980,317,488,280,984,252,706,743,454,967,247,070组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{243}\frac{x^{366}-x^i}{x^{i+122}-x^{122}},(x,0,14824)\bigg],x\bigg]\)[[14824]]

从1~365中取132个不同数, 132个数之和恰好等于24156有 953,163,733,503,859,111,013,051,616,701,688,309,768,626,876,182,270,382,208,477,563,894,791,704,733,584,473,982,505,746,826,001,976组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{233}\frac{x^{366}-x^i}{x^{i+117}-x^{117}},(x,0,15379)\bigg],x\bigg]\)[[15379]]

从1~365中取142个不同数, 142个数之和恰好等于25986有 151,111,644,158,877,461,615,176,016,165,544,111,064,863,470,331,198,744,416,850,124,028,633,790,549,084,960,148,265,478,359,884,017,320组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{223}\frac{x^{366}-x^i}{x^{i+112}-x^{112}},(x,0,15834)\bigg],x\bigg]\)[[15834]]

从1~365中取152个不同数, 152个数之和恰好等于27816有 7,612,758,755,330,028,426,971,377,342,459,599,737,864,438,968,185,822,628,924,554,168,720,288,895,463,943,491,508,406,071,720,893,135,890组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{213}\frac{x^{366}-x^i}{x^{i+107}-x^{107}},(x,0,16189)\bigg],x\bigg]\)[[16189]]

从1~365中取162个不同数, 162个数之和恰好等于29646有 124,946,702,870,321,967,235,134,974,592,997,204,874,275,672,831,335,078,368,012,511,567,260,443,497,143,702,125,544,961,809,200,467,128,672组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{203}\frac{x^{366}-x^i}{x^{i+102}-x^{102}},(x,0,16444)\bigg],x\bigg]\)[[16444]]

从1~365中取172个不同数, 172个数之和恰好等于31476有 679,728,004,010,725,708,247,275,762,250,148,714,810,033,837,556,572,821,688,774,046,128,985,369,426,971,236,533,144,606,302,636,446,352,536组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{193}\frac{x^{366}-x^i}{x^{i+97}-x^{97}},(x,0,16599)\bigg],x\bigg]\)[[16599]]

从1~365中取182个不同数, 182个数之和恰好等于33306有 1,238,265,005,128,257,960,854,030,011,771,898,020,954,511,596,775,351,614,116,958,454,583,975,724,255,436,480,191,559,141,334,272,573,713,844组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{183}\frac{x^{366}-x^i}{x^{i+92}-x^{92}},(x,0,16654)\bigg],x\bigg]\)[[16654]]

从1~365中取192个不同数, 192个数之和恰好等于35136有 758,081,700,107,993,035,974,870,570,958,151,684,770,830,178,845,299,115,482,249,293,474,685,990,743,934,031,053,607,491,130,986,299,383,766组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{173}\frac{x^{366}-x^i}{x^{i+87}-x^{87}},(x,0,16609)\bigg],x\bigg]\)[[16609]]

从1~365中取202个不同数, 202个数之和恰好等于36966有 155,514,236,933,631,890,218,139,215,530,080,854,070,079,724,023,883,542,506,428,889,829,152,951,912,499,151,127,994,560,155,297,814,827,366组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{163}\frac{x^{366}-x^i}{x^{i+82}-x^{82}},(x,0,16464)\bigg],x\bigg]\)[[16464]]

从1~365中取212个不同数, 212个数之和恰好等于38796有 10,588,380,801,539,873,124,471,285,828,821,229,065,127,060,067,905,073,897,150,999,733,500,302,048,969,024,466,486,639,266,713,969,943,140组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{153}\frac{x^{366}-x^i}{x^{i+77}-x^{77}},(x,0,16219)\bigg],x\bigg]\)[[16219]]

从1~365中取222个不同数, 222个数之和恰好等于40626有 235,353,548,961,557,518,090,265,855,544,362,566,381,076,303,491,753,610,150,198,803,572,512,308,695,326,680,782,939,221,354,363,049,912组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{143}\frac{x^{366}-x^i}{x^{i+72}-x^{72}},(x,0,15874)\bigg],x\bigg]\)[[15874]]

从1~365中取232个不同数, 232个数之和恰好等于42456有 1,667,129,803,153,419,350,812,012,857,889,233,121,847,924,396,907,857,590,349,580,906,332,765,313,443,023,086,451,903,964,492,837,064组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{133}\frac{x^{366}-x^i}{x^{i+67}-x^{67}},(x,0,15429)\bigg],x\bigg]\)[[15429]]

从1~365中取242个不同数, 242个数之和恰好等于44286有 3,642,319,656,811,713,231,479,712,181,723,377,007,094,138,968,197,459,573,160,046,031,199,900,221,975,544,688,935,092,012,191,512组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{123}\frac{x^{366}-x^i}{x^{i+62}-x^{62}},(x,0,14884)\bigg],x\bigg]\)[[14884]]

从1~365中取252个不同数, 252个数之和恰好等于46116有 2,351,852,253,191,729,120,650,288,329,465,613,792,778,715,770,215,355,092,264,103,255,736,054,761,882,201,199,657,550,173,512组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{113}\frac{x^{366}-x^i}{x^{i+57}-x^{57}},(x,0,14239)\bigg],x\bigg]\)[[14239]]

从1~365中取262个不同数, 262个数之和恰好等于47946有 424,919,401,107,490,493,518,326,367,912,045,983,124,343,990,522,312,515,326,340,333,333,114,957,222,134,687,397,253,984组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{103}\frac{x^{366}-x^i}{x^{i+52}-x^{52}},(x,0,13494)\bigg],x\bigg]\)[[13494]]

从1~365中取272个不同数, 272个数之和恰好等于49776有 20,036,313,201,925,393,250,559,788,653,853,214,948,002,538,959,790,992,550,434,572,106,534,584,361,520,820,599,658组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{93}\frac{x^{366}-x^i}{x^{i+47}-x^{47}},(x,0,12649)\bigg],x\bigg]\)[[12649]]

从1~365中取282个不同数, 282个数之和恰好等于51606有 225,605,670,846,057,198,414,201,222,180,316,675,053,629,232,373,832,083,827,237,812,946,676,964,835,637,830组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{83}\frac{x^{366}-x^i}{x^{i+42}-x^{42}},(x,0,11704)\bigg],x\bigg]\)[[11704]]

从1~365中取292个不同数, 292个数之和恰好等于53436有 540,984,660,192,891,612,140,640,600,117,334,181,790,325,017,800,698,031,840,796,058,589,909,220,173组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{73}\frac{x^{366}-x^i}{x^{i+37}-x^{37}},(x,0,10659)\bigg],x\bigg]\)[[10659]]

从1~365中取302个不同数, 302个数之和恰好等于55266有 237,762,695,973,447,690,351,046,491,893,641,153,195,144,866,856,706,230,388,026,059,555,530组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{63}\frac{x^{366}-x^i}{x^{i+32}-x^{32}},(x,0,9514)\bigg],x\bigg]\)[[9514]]

从1~365中取312个不同数, 312个数之和恰好等于57096有 15,648,184,566,422,670,867,196,470,627,366,929,481,481,374,672,851,806,953,546,944组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{53}\frac{x^{366}-x^i}{x^{i+27}-x^{27}},(x,0,8269)\bigg],x\bigg]\)[[8269]]

从1~365中取322个不同数, 322个数之和恰好等于58926有 116,212,775,733,205,156,816,065,702,872,450,113,210,545,218,239,534,542组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{43}\frac{x^{366}-x^i}{x^{i+22}-x^{22}},(x,0,6924)\bigg],x\bigg]\)[[6924]]

从1~365中取332个不同数, 332个数之和恰好等于60756有 63,995,334,301,723,685,441,535,021,060,219,672,927,820,493组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{33}\frac{x^{366}-x^i}{x^{i+17}-x^{17}},(x,0,5479)\bigg],x\bigg]\)[[5479]]

从1~365中取342个不同数, 342个数之和恰好等于62886有 1,320,827,419,936,988,491,388,265,553,443,322组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{23}\frac{x^{366}-x^i}{x^{i+12}-x^{12}},(x,0,3934)\bigg],x\bigg]\)[[3934]]

从1~365中取352个不同数,352个数之和恰好等于64416有 278,664,877,516,110,742,675组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{13}\frac{x^{366}-x^i}{x^{i+7}-x^{7}},(x,0,2289)\bigg],x\bigg]\)[[2289]]

从1~365中取362个不同数,362个数之和恰好等于66246有 16,562组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{3}\frac{x^{366}-x^i}{x^{i+2}-x^{2}},(x,0,544)\bigg],x\bigg]\)[[544]]

王守恩

谢谢 northwolves！！！这是2017年的题目，当时把我吓着了。

从1到93这93个自然数中任取两两不等的9个数, 把它们相加得到一个和;

S(45)=S(801)=1
S(46)=S(800)=1
S(47)=S(799)=2
S(48)=S(798)=3
S(49)=S(797)=5
S(50)=S(796)=7
S(51)=S(795)=11
S(52)=S(794)=15
S(53)=S(793)=22
S(54)=S(792)=30
S(55)=S(791)=41
S(56)=S(790)=54
S(57)=S(789)=73
S(58)=S(788)=94
S(59)=S(787)=123
S(60)=S(786)=157
S(61)=S(785)=201
S(62)=S(784)=252
S(63)=S(783)=318
S(64)=S(782)=393

CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{84}\frac{x^{94}-x^i}{x^{i+85/2}-x^{85/2}},(x,0,756)\bigg],x\bigg]\)

{1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 94, 123, 157, 201, 252, 318, 393, 488, 598, 732, 887, 1076,1291, 1549, 1845, 2194, 2592, 3060, 3589, 4206, 4904, 5708, 6615, 7657, 8824,
10156, 11648, 13338, 15224, 17354, 19720, 22380, 25331, 28629, 32278, 36347, 40831, 45812, 51294, 57358,64015, 71362, 79403, 88252, 97922, 108527, 120092, 132751, 146520,
161554, 177884, 195666, 214944, 235899, 258569, 283161, 309729, 338484, 369499, 403016, 439100, 478025, 519880, 564945, 613331, 665355, 721125, 780997, 845105, 913815,
987292, 1065947, 1149940, 1239730, 1335513, 1437762, 1546707, 1662875, 1786498, 1918159, 2058130, 2207014, 2365128, 2533138, 2711363, 2900536, 3101027, 3313594, 3538657,
3777049, 4029199, 4296017, 4577988, 4876054, 5190763, 5523142, 5873748, 6243698, 6633618, 7044663, 7477526, 7933460, 8413176, 8918027, 9448798, 10006888,  10593163,  
11209130, 11855672, 12534410, 13246314, 13993056, 14775691, 15596013,16455101,17354873, 18296499, 19281955, 20312509, 21390268, 22516522, 23693514, 24922637, 26206195,
27545681, 28943542, 30401298, 31921540, 33505891, 35157009, 36876627,38667552,40531537, 42471543, 44489436, 46588241, 48769932, 51037692, 53393516, 55840744, 58381479,
61019126, 63755901, 66595368, 69539751, 72592776, 75756778, 79035540, 82431501, 85948605, 89589293, 93357667, 97256267, 101289245, 105459243, 109770567, 114225838,
118829517, 123584320, 128494739, 133563574, 138795465, 144193176, 149761487, 155503235, 161423215, 167524336, 173811523, 180287616, 186957668, 193824577, 200893381,
208167022, 215650651, 223347116, 231261670, 239397187, 247758877, 256349637, 265174762, 274237010, 283541755, 293091757, 302892307, 312946148, 323258630, 333832322,
344672617, 355782044, 367165875, 378826593, 390769489, 402996819, 415513886,428322873, 441428912, 454834092, 468543531, 482559047, 496885726, 511525264, 526482531,
541759093, 557359760,573285767, 589541854, 606129097, 623051964, 640311348, 657911619, 675853292, 694140619, 712773902, 731757073, 751090214, 770777110, 790817402,
811214719, 831968452, 853081852, 874554037, 896388074, 918582594, 941140461, 964060005, 987343667, 1010989469, 1034999623,1059371606,1084107393,1109204129,1134663313,
1160481737, 1186660641, 1213196234, 1240089481, 1267336216, 1294936893, 1322886970, 1351186608, 1379830634, 1408818914, 1438145884, 1467810859, 1497807868, 1528135918,
1558788388, 1589763966, 1621055615, 1652661456, 1684574041, 1716791167, 1749304709, 1782112145, 1815204938, 1848579981, 1882228317, 1916146522, 1950324966, 1984759905,
2019441296, 2054364816, 2089520024, 2124902287, 2160500490, 2196309704, 2232318431, 2268521171, 2304906043, 2341467270, 2378192329, 2415075170, 2452102912, 2489268968,
2526560123, 2563969544, 2601483400, 2639094636, 2676789120, 2714559293, 2752390731, 2790275688, 2828199183, 2866153296, 2904122796, 2942099318, 2980067414, 3018018588,
3055936887, 3093813715, 3131632951, 3169385612, 3207055425, 3244633357, 3282102714, 3319454433, 3356671724, 3393745219, 3430658073, 3467400943, 3503956636, 3540315872,
3576461465, 3612383904, 3648066032, 3683498462, 3718663790, 3753552777, 3788148113, 3822440426, 3856412545, 3890055309, 3923351387, 3956291870, 3988859635, 4021045724,
4052833246, 4084213559, 4115169723, 4145693438, 4175768064, 4205385354, 4234529015, 4263191205, 4291355669, 4319015011, 4346153390, 4372763545, 4398830071, 4424346213,
4449296711, 4473675337, 4497467330, 4520666691, 4543259199, 4565239437, 4586593406, 4607316307, 4627394737, 4646824193, 4665591885, 4683693975, 4701117983, 4717860750,
4733910457, 4749264314, 4763911196, 4777849031, 4791067057, 4803563947, 4815329672, 4826363315, 4836655586, 4846206342, 4855006717, 4863057344, 4870350125, 4876886147,
4882658102, 4887667871, 4891908593, 4895382960, 4898084917, 4900017619, 4901175813, 4901563471, 4901175813, 4900017619, 4898084917, 4895382960, 4891908593, 4887667871,
4882658102, 4876886147, 4870350125, 4863057344, 4855006717, 4846206342, 4836655586, 4826363315, 4815329672, 4803563947, 4791067057, 4777849031, 4763911196, 4749264314,
4733910457, 4717860750, 4701117983, 4683693975, 4665591885, 4646824193, 4627394737, 4607316307, 4586593406, 4565239437, 4543259199, 4520666691, 4497467330, 4473675337,
4449296711, 4424346213, 4398830071, 4372763545, 4346153390, 4319015011, 4291355669, 4263191205, 4234529015, 4205385354, 4175768064, 4145693438, 4115169723, 4084213559,
4052833246, 4021045724, 3988859635, 3956291870, 3923351387, 3890055309, 3856412545, 3822440426, 3788148113, 3753552777, 3718663790, 3683498462, 3648066032, 3612383904,
3576461465, 3540315872, 3503956636, 3467400943, 3430658073, 3393745219, 3356671724, 3319454433, 3282102714, 3244633357, 3207055425, 3169385612, 3131632951, 3093813715,
3055936887, 3018018588, 2980067414, 2942099318, 2904122796, 2866153296, 2828199183, 2790275688, 2752390731, 2714559293, 2676789120, 2639094636, 2601483400, 2563969544,
2526560123, 2489268968, 2452102912, 2415075170, 2378192329, 2341467270, 2304906043, 2268521171, 2232318431, 2196309704, 2160500490, 2124902287, 2089520024, 2054364816,
2019441296, 1984759905, 1950324966, 1916146522, 1882228317, 1848579981, 1815204938, 1782112145, 1749304709, 1716791167, 1684574041, 1652661456, 1621055615, 1589763966,
1558788388, 1528135918, 1497807868, 1467810859, 1438145884, 1408818914, 1379830634, 1351186608, 1322886970, 1294936893, 1267336216, 1240089481, 1213196234, 1186660641,
1160481737, 1134663313, 1109204129, 1084107393, 1059371606, 1034999623, 1010989469,  987343667,   964060005,   941140461,918582594, 896388074, 874554037, 853081852,
831968452, 811214719, 790817402, 770777110, 751090214, 731757073, 712773902, 694140619, 675853292, 657911619, 640311348, 623051964,606129097, 589541854, 573285767,
557359760, 541759093, 526482531, 511525264, 496885726,482559047, 468543531, 454834092, 441428912, 428322873, 415513886, 402996819, 390769489, 378826593, 367165875,
355782044, 344672617, 333832322, 323258630, 312946148, 302892307,293091757, 283541755, 274237010, 265174762, 256349637, 247758877, 239397187, 231261670,223347116,
215650651, 208167022, 200893381, 193824577, 186957668, 180287616, 173811523,167524336, 161423215, 155503235, 149761487, 144193176, 138795465, 133563574, 128494739,
123584320, 118829517, 114225838, 109770567, 105459243, 101289245, 97256267, 93357667, 89589293, 85948605, 82431501, 79035540, 75756778, 72592776, 69539751, 66595368,
63755901,61019126, 58381479, 55840744, 53393516, 51037692,48769932,46588241, 44489436, 42471543, 40531537, 38667552, 36876627, 35157009, 33505891, 31921540, 30401298,
28943542, 27545681,26206195, 24922637, 23693514, 22516522, 21390268, 20312509,19281955, 18296499, 17354873, 16455101, 15596013, 14775691, 13993056, 13246314, 12534410,
11855672, 11209130, 10593163,10006888, 9448798, 8918027, 8413176,7933460, 7477526, 7044663, 6633618, 6243698,5873748,5523142,5190763,4876054,4577988, 4296017,4029199,
3777049, 3538657, 3313594, 3101027, 2900536, 2711363, 2533138, 2365128, 2207014, 2058130, 1918159, 1786498, 1662875, 1546707,1437762, 1335513, 1239730, 1149940, 1065947,
987292, 913815, 845105,780997,721125,665355,613331, 564945, 519880, 478025, 439100, 403016, 369499,338484, 309729, 283161, 258569, 235899, 214944, 195666, 177884, 161554,
146520, 132751, 120092, 108527, 97922, 88252, 79403, 71362, 64015, 57358, 51294, 45812, 40831, 36347, 32278,28629,25331, 22380, 19720,17354, 15224, 13338,11648, 10156, 8824,
7657, 6615, 5708, 4904, 4206, 3589, 3060, 2592, 2194, 1845, 1549, 1291, 1076, 887, 732, 598, 488, 393, 318, 252, 201, 157, 123, 94, 73, 54, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1}

王守恩

谢谢 northwolves！利用21楼的公式。

从1到93这93个自然数中任取两两不等的9个数, 把它们相加得到一个和;

根据模2
   {961835834245, {0 -> 480917835530, 1 -> 480917998715}}

根据模3
   {961835834245, {0 -> 320611969669, 1 -> 320611932288, 2 -> 320611932288}}

根据模4
   {961835834245, {0 -> 240458928561, 1 -> 240459018024, 2 -> 240458906969, 3 -> 240458980691}}

根据模5
   {961835834245, {0 -> 192367166849, 1 -> 192367166849, 2 -> 192367166849, 3 -> 192367166849, 4 -> 192367166849}}

根据模6
   {961835834245, {0 -> 160305833348, 1 -> 160306172766, 2 -> 160306242660, 3 -> 160306136321, 4 -> 160305759522, 5 -> 160305689628}}

根据模7
   {961835834245, {0 -> 137404699655, 1 -> 137405071093, 2 -> 137405480090, 3 -> 137405615242, 4 -> 137405378330,  5 -> 137404945807, 6 -> 137404644028}}

根据模8
   {961835834245, {0 -> 120229471007, 1 -> 120229496813, 2 -> 120229429363, 3 -> 120229467724, 4 -> 120229457554, 5 -> 120229521211, 6 -> 120229477606, 7 -> 120229512967}}

根据模9
   {961835834245, {0 -> 106870656520, 1 -> 106870644090, 2 -> 106870644090, 3 -> 106870656555, 4 -> 106870644117,
    5 -> 106870644117, 6 -> 106870656594, 7 -> 106870644081,  8 -> 106870644081}}

根据模10
   {961835834245, {0 -> 96183567106, 1 -> 96183599743, 2 -> 96183567106, 3 -> 96183599743, 4 -> 96183567106,
    5 -> 96183599743, 6 -> 96183567106,7 -> 96183599743, 8 -> 96183567106, 9 -> 96183599743}}

根据模11
   {961835834245, {0 -> 87439624480, 1 -> 87439621477, 2 -> 87439597167, 3 -> 87439621477, 4 -> 87439634347,
    5 -> 87439634347, 6 -> 87439634347, 7 -> 87439615042, 8 -> 87439615042, 9 -> 87439615042, 10 -> 87439621477}}

根据模12
   {961835834245, {0 -> 80152963488, 1 -> 80153371017, 2 -> 80153556426,  3 -> 80153543621, 4 -> 80153278839, 5 -> 80153054307,
    6 -> 80152869860, 7 -> 80152801749, 8 -> 80152686234, 9 -> 80152592700, 10 -> 80152480683, 11 -> 80152635321}}

根据模13
   {961835834245, {0 -> 73993597309, 1 -> 73992716506, 2 -> 73990737364, 3 -> 73988166730, 4 -> 73985592279, 5 -> 73983322066, 6 -> 73981828354,
    7 -> 73981380952, 8 -> 73982359830,  9 -> 73984624114,10 -> 73987750534, 11 -> 73990809535, 12 -> 73992948672}}

根据模14
   {961835834245, {0 -> 68713636855, 1 -> 68698889469, 2 -> 68684857015, 3 -> 68674272652, 4 -> 68669092888, 5 -> 68670535932, 6 -> 68678314683,
    7 -> 68691062800, 8 -> 68706181624,  9 -> 68720623075, 10 -> 68731342590,11 -> 68736285442, 12 -> 68734409875, 13 -> 68726329345}}

根据模15
   {961835834245, {0 -> 64039372505, 1 -> 64013017320, 2 -> 64005644223, 3 -> 64018367801, 4 -> 64049154318, 5 -> 64092579840, 6 -> 64141264565,
    7 -> 64186635357, 8 -> 64220888259, 9 -> 64238077529, 10 -> 64235214504, 11 -> 64212884964, 12 -> 64174887269, 13 -> 64127910789, 14 -> 64079935002}}
......

其中余数各不相同的是模2,4,5,6,7,8,12,13,14,15,16,17,18,19,20,21,22,23,......好像没有规律？

25楼：这串数已经有了，就不能按模分一下？

northwolves

1. Table[{n, Round[n/3], Round[n/3], n - 2 Round[n/3]}, {n, 11, 30}]

复制代码

{{11,4,4,3},{12,4,4,4},{13,4,4,5},{14,5,5,4},{15,5,5,5},{16,5,5,6},{17,6,6,5},{18,6,6,6},{19,6,6,7},{20,7,7,6},{21,7,7,7},{22,7,7,8},{23,8,8,7},{24,8,8,8},{25,8,8,9},{26,9,9,8},{27,9,9,9},{28,9,9,10},{29,10,10,9},{30,10,10,10}}

northwolves

或者  

1. Table[{n, Last@IntegerPartitions[n, {3}]}, {n, 11, 30}]

复制代码

{{11,{4,4,3}},{12,{4,4,4}},{13,{5,4,4}},{14,{5,5,4}},{15,{5,5,5}},{16,{6,5,5}},{17,{6,6,5}},{18,{6,6,6}},{19,{7,6,6}},{20,{7,7,6}},{21,{7,7,7}},{22,{8,7,7}},{23,{8,8,7}},{24,{8,8,8}},{25,{9,8,8}},{26,{9,9,8}},{27,{9,9,9}},{28,{10,9,9}},{29,{10,10,9}},{30,{10,10,10}}}

王守恩

northwolves 发表于 2023-5-30 17:57
或者   

{{11,{4,4,3}},{12,{4,4,4}},{13,{5,4,4}},{14,{5,5,4}},{15,{5,5,5}},{16,{6,5,5}},{17,{6,6,5 ...

加2个字，还可以吗？
3条边(a,b,c)都是整数(不同)的三角形，我们取面积最大的,
当a+b+c=n,  n=11,12,13,14,15,16,17,18, ......
请您写出用n来表示3条边的通项公式。

northwolves

王守恩 发表于 2023-5-30 18:24
加2个字，还可以吗？
3条边(a,b,c)都是整数(不同)的三角形，我们取面积最大的,
当a+b+c=n,  n=11,12,13 ...

1. Table[{n,
   
2.   Last@Select[IntegerPartitions[n, {3}],
   
3.     CountDistinct[#] == 3 &]}, {n, 11, 30}]

复制代码

{{11,{5,4,2}},{12,{5,4,3}},{13,{6,4,3}},{14,{6,5,3}},{15,{6,5,4}},{16,{7,5,4}},{17,{7,6,4}},{18,{7,6,5}},{19,{8,6,5}},{20,{8,7,5}},{21,{8,7,6}},{22,{9,7,6}},{23,{9,8,6}},{24,{9,8,7}},{25,{10,8,7}},{26,{10,9,7}},{27,{10,9,8}},{28,{11,9,8}},{29,{11,10,8}},{30,{11,10,9}}}

王守恩

northwolves 发表于 2023-5-30 23:14
{{11,{5,4,2}},{12,{5,4,3}},{13,{6,4,3}},{14,{6,5,3}},{15,{6,5,4}},{16,{7,5,4}},{17,{7,6,4}},{1 ...

谢谢northwolves ！在我这里高大难的问题，怎么到你那里就变成小儿科了？佩服佩服！

根据a,b,c3个数，我们可以有面积=\(\frac{\sqrt{(a + b + c) (c + a - b) (b + c - a) (a + b - c)}}{4}\)

我们只取\((a + b + c) (c + a - b) (b + c - a) (a + b - c),\)得到这样一串数：

231,576,455,896,1575,1536,2295,3456,3591,4800,6615,7040,8855,11520,12375,14976,18711,20160,......

怎样会把这串数单独提出来？谢谢！

nyy

整天研究这类没啥意义的问题？？？？？？？

northwolves

王守恩 发表于 2023-5-31 10:51
谢谢northwolves ！在我这里高大难的问题，怎么到你那里就变成小儿科了？佩服佩服！

根据a,b,c3个数， ...

1. Total[#]*(Total[#] - 2 #[[1]]) (Total[#] - 2 #[[2]]) (Total[#] -
   
2.      2 #[[3]]) & /@
   
3. Table[Last@
   
4.    Select[IntegerPartitions[n, {3}], CountDistinct[#] == 3 &], {n, 11,
   
5.     30}]

复制代码

王守恩

northwolves 发表于 2023-5-31 13:29

谢谢northwolves ！8楼是这样一串数：

231,576,455,896,1575,1536,2295,3456,3591,4800,6615,7040,8855,11520,12375,14976,18711,20160,......

这串数感觉太大了，若我们只取：\((c+a-b)(b+c-a)(a+b-c)\)

怎样把这串数提出来？我真不会用这些按钮(继续学习)。谢谢northwolves ！

王守恩

nyy 发表于 2023-5-31 11:49
整天研究这类没啥意义的问题？？？？？？？

主帖想解决这样一个问题。

3条边都是整数(不同)的三角形，已知1条边与周长, 求三角形的最大面积。