煎饼数 蛋糕数——有趣的自然数及其性质（四）

yangchuanju

煎饼数
pancake numbers
https://www.numbersaplenty.com/set/pancake_number/

Pancake number Pn represents the maximal number of pieces in which a pancake (or a circle) can be divided into by n linear cuts. They are the bidimensional version of cake numbers.
煎饼数Pn表示煎饼（或圆形）可以被n条直线切割分成的最大块数。它们是蛋糕数字的二维版本。

In general, 一般来说，
Pn=C(n+1,2)+1=C(n,2)+C(n,1)+C(n,0)=(n^2+n+2)/2

Pancake numbers are often called lazy caterer numbers, or, more formally, central polygonal numbers.
煎饼数通常称为惰性餐饮服务数，或更正式地称为中心多边形数。

The first pancake numbers are  最初的一些煎饼数是
1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326 more terms

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The first 600 pancake numbers :前600个煎饼数是：
1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432, 1486, 1541, 1597, 1654, 1712, 1771, 1831, 1892, 1954, 2017, 2081, 2146, 2212, 2279, 2347, 2416, 2486, 2557, 2629, 2702, 2776, 2851, 2927, 3004, 3082, 3161, 3241, 3322, 3404, 3487, 3571, 3656, 3742, 3829, 3917, 4006, 4096, 4187, 4279, 4372, 4466, 4561, 4657, 4754, 4852, 4951, 5051, 5152, 5254, 5357, 5461, 5566, 5672, 5779, 5887, 5996, 6106, 6217, 6329, 6442, 6556, 6671, 6787, 6904, 7022, 7141, 7261, 7382, 7504, 7627, 7751, 7876, 8002, 8129, 8257, 8386, 8516, 8647, 8779, 8912, 9046, 9181, 9317, 9454, 9592, 9731, 9871, 10012, 10154, 10297, 10441, 10586, 10732, 10879, 11027, 11176, 11326, 11477, 11629, 11782, 11936, 12091, 12247, 12404, 12562, 12721, 12881, 13042, 13204, 13367, 13531, 13696, 13862, 14029, 14197, 14366, 14536, 14707, 14879, 15052, 15226, 15401, 15577, 15754, 15932, 16111, 16291, 16472, 16654, 16837, 17021, 17206, 17392, 17579, 17767, 17956, 18146, 18337, 18529, 18722, 18916, 19111, 19307, 19504, 19702, 19901, 20101, 20302, 20504, 20707, 20911, 21116, 21322, 21529, 21737, 21946, 22156, 22367, 22579, 22792, 23006, 23221, 23437, 23654, 23872, 24091, 24311, 24532, 24754, 24977, 25201, 25426, 25652, 25879, 26107, 26336, 26566, 26797, 27029, 27262, 27496, 27731, 27967, 28204, 28442, 28681, 28921, 29162, 29404, 29647, 29891, 30136, 30382, 30629, 30877, 31126, 31376, 31627, 31879, 32132, 32386, 32641, 32897, 33154, 33412, 33671, 33931, 34192, 34454, 34717, 34981, 35246, 35512, 35779, 36047, 36316, 36586, 36857, 37129, 37402, 37676, 37951, 38227, 38504, 38782, 39061, 39341, 39622, 39904, 40187, 40471, 40756, 41042, 41329, 41617, 41906, 42196, 42487, 42779, 43072, 43366, 43661, 43957, 44254, 44552, 44851, 45151, 45452, 45754, 46057, 46361, 46666, 46972, 47279, 47587, 47896, 48206, 48517, 48829, 49142, 49456, 49771, 50087, 50404, 50722, 51041, 51361, 51682, 52004, 52327, 52651, 52976, 53302, 53629, 53957, 54286, 54616, 54947, 55279, 55612, 55946, 56281, 56617, 56954, 57292, 57631, 57971, 58312, 58654, 58997, 59341, 59686, 60032, 60379, 60727, 61076, 61426, 61777, 62129, 62482, 62836, 63191, 63547, 63904, 64262, 64621, 64981, 65342, 65704, 66067, 66431, 66796, 67162, 67529, 67897, 68266, 68636, 69007, 69379, 69752, 70126, 70501, 70877, 71254, 71632, 72011, 72391, 72772, 73154, 73537, 73921, 74306, 74692, 75079, 75467, 75856, 76246, 76637, 77029, 77422, 77816, 78211, 78607, 79004, 79402, 79801, 80201, 80602, 81004, 81407, 81811, 82216, 82622, 83029, 83437, 83846, 84256, 84667, 85079, 85492, 85906, 86321, 86737, 87154, 87572, 87991, 88411, 88832, 89254, 89677, 90101, 90526, 90952, 91379, 91807, 92236, 92666, 93097, 93529, 93962, 94396, 94831, 95267, 95704, 96142, 96581, 97021, 97462, 97904, 98347, 98791, 99236, 99682, 100129, 100577, 101026, 101476, 101927, 102379, 102832, 103286, 103741, 104197, 104654, 105112, 105571, 106031, 106492, 106954, 107417, 107881, 108346, 108812, 109279, 109747, 110216, 110686, 111157, 111629, 112102, 112576, 113051, 113527, 114004, 114482, 114961, 115441, 115922, 116404, 116887, 117371, 117856, 118342, 118829, 119317, 119806, 120296, 120787, 121279, 121772, 122266, 122761, 123257, 123754, 124252, 124751, 125251, 125752, 126254, 126757, 127261, 127766, 128272, 128779, 129287, 129796, 130306, 130817, 131329, 131842, 132356, 132871, 133387, 133904, 134422, 134941, 135461, 135982, 136504, 137027, 137551, 138076, 138602, 139129, 139657, 140186, 140716, 141247, 141779, 142312, 142846, 143381, 143917, 144454, 144992, 145531, 146071, 146612, 147154, 147697, 148241, 148786, 149332, 149879, 150427, 150976, 151526, 152077, 152629, 153182, 153736, 154291, 154847, 155404, 155962, 156521, 157081, 157642, 158204, 158767, 159331, 159896, 160462, 161029, 161597, 162166, 162736, 163307, 163879, 164452, 165026, 165601, 166177, 166754, 167332, 167911, 168491, 169072, 169654, 170237, 170821, 171406, 171992, 172579, 173167, 173756, 174346, 174937, 175529, 176122, 176716, 177311, 177907, 178504, 179102, 179701.

另A000124给出1000个煎饼数。

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蛋糕数
cake numbers
https://www.numbersaplenty.com/set/cake_number/

Cake number Cn represents the maximal number of pieces in which a cake (or a cube) can be divided into by n planar cuts. They are the tridimensional version of pancake numbers.
蛋糕数Cn表示蛋糕（或立方体）可以通过n个平面切割分成的最大块数。它们是煎饼数的三维版本。

In general, 一般来说，
Cn=C(n+1,3)+n+1=C(n,3)+C(n,2)+C(n,1)+C(n,0)=(n^3+5*n+6)/6

Note that with the first 3 cuts it is possible to divide the cake into 2, then 4 and finally 8 pieces, but it is impossible to cut each of the 8 pieces with a fourth cut, and indeed     is only 15, not 16.
请注意，前 3 次切割可以将蛋糕分成 2 块，然后是 4 块，最后是 8 块，但不可能用第四次切割将 8 块中的每块都切开，而且实际上只有 15 块，而不是 16块。

The first cake numbers are 最初的一些蛋糕数是1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807 more terms

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The first 600 cake numbers :前600个蛋糕数是：
1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235, 15226, 16262, 17344, 18473, 19650, 20876, 22152, 23479, 24858, 26290, 27776, 29317, 30914, 32568, 34280, 36051, 37882, 39774, 41728, 43745, 45826, 47972, 50184, 52463, 54810, 57226, 59712, 62269, 64898, 67600, 70376, 73227, 76154, 79158, 82240, 85401, 88642, 91964, 95368, 98855, 102426, 106082, 109824, 113653, 117570, 121576, 125672, 129859, 134138, 138510, 142976, 147537, 152194, 156948, 161800, 166751, 171802, 176954, 182208, 187565, 193026, 198592, 204264, 210043, 215930, 221926, 228032, 234249, 240578, 247020, 253576, 260247, 267034, 273938, 280960, 288101, 295362, 302744, 310248, 317875, 325626, 333502, 341504, 349633, 357890, 366276, 374792, 383439, 392218, 401130, 410176, 419357, 428674, 438128, 447720, 457451, 467322, 477334, 487488, 497785, 508226, 518812, 529544, 540423, 551450, 562626, 573952, 585429, 597058, 608840, 620776, 632867, 645114, 657518, 670080, 682801, 695682, 708724, 721928, 735295, 748826, 762522, 776384, 790413, 804610, 818976, 833512, 848219, 863098, 878150, 893376, 908777, 924354, 940108, 956040, 972151, 988442, 1004914, 1021568, 1038405, 1055426, 1072632, 1090024, 1107603, 1125370, 1143326, 1161472, 1179809, 1198338, 1217060, 1235976, 1255087, 1274394, 1293898, 1313600, 1333501, 1353602, 1373904, 1394408, 1415115, 1436026, 1457142, 1478464, 1499993, 1521730, 1543676, 1565832, 1588199, 1610778, 1633570, 1656576, 1679797, 1703234, 1726888, 1750760, 1774851, 1799162, 1823694, 1848448, 1873425, 1898626, 1924052, 1949704, 1975583, 2001690, 2028026, 2054592, 2081389, 2108418, 2135680, 2163176, 2190907, 2218874, 2247078, 2275520, 2304201, 2333122, 2362284, 2391688, 2421335, 2451226, 2481362, 2511744, 2542373, 2573250, 2604376, 2635752, 2667379, 2699258, 2731390, 2763776, 2796417, 2829314, 2862468, 2895880, 2929551, 2963482, 2997674, 3032128, 3066845, 3101826, 3137072, 3172584, 3208363, 3244410, 3280726, 3317312, 3354169, 3391298, 3428700, 3466376, 3504327, 3542554, 3581058, 3619840, 3658901, 3698242, 3737864, 3777768, 3817955, 3858426, 3899182, 3940224, 3981553, 4023170, 4065076, 4107272, 4149759, 4192538, 4235610, 4278976, 4322637, 4366594, 4410848, 4455400, 4500251, 4545402, 4590854, 4636608, 4682665, 4729026, 4775692, 4822664, 4869943, 4917530, 4965426, 5013632, 5062149, 5110978, 5160120, 5209576, 5259347, 5309434, 5359838, 5410560, 5461601, 5512962, 5564644, 5616648, 5668975, 5721626, 5774602, 5827904, 5881533, 5935490, 5989776, 6044392, 6099339, 6154618, 6210230, 6266176, 6322457, 6379074, 6436028, 6493320, 6550951, 6608922, 6667234, 6725888, 6784885, 6844226, 6903912, 6963944, 7024323, 7085050, 7146126, 7207552, 7269329, 7331458, 7393940, 7456776, 7519967, 7583514, 7647418, 7711680, 7776301, 7841282, 7906624, 7972328, 8038395, 8104826, 8171622, 8238784, 8306313, 8374210, 8442476, 8511112, 8580119, 8649498, 8719250, 8789376, 8859877, 8930754, 9002008, 9073640, 9145651, 9218042, 9290814, 9363968, 9437505, 9511426, 9585732, 9660424, 9735503, 9810970, 9886826, 9963072, 10039709, 10116738, 10194160, 10271976, 10350187, 10428794, 10507798, 10587200, 10667001, 10747202, 10827804, 10908808, 10990215, 11072026, 11154242, 11236864, 11319893, 11403330, 11487176, 11571432, 11656099, 11741178, 11826670, 11912576, 11998897, 12085634, 12172788, 12260360, 12348351, 12436762, 12525594, 12614848, 12704525, 12794626, 12885152, 12976104, 13067483, 13159290, 13251526, 13344192, 13437289, 13530818, 13624780, 13719176, 13814007, 13909274, 14004978, 14101120, 14197701, 14294722, 14392184, 14490088, 14588435, 14687226, 14786462, 14886144, 14986273, 15086850, 15187876, 15289352, 15391279, 15493658, 15596490, 15699776, 15803517, 15907714, 16012368, 16117480, 16223051, 16329082, 16435574, 16542528, 16649945, 16757826, 16866172, 16974984, 17084263, 17194010, 17304226, 17414912, 17526069, 17637698, 17749800, 17862376, 17975427, 18088954, 18202958, 18317440, 18432401, 18547842, 18663764, 18780168, 18897055, 19014426, 19132282, 19250624, 19369453, 19488770, 19608576, 19728872, 19849659, 19970938, 20092710, 20214976, 20337737, 20460994, 20584748, 20709000, 20833751, 20959002, 21084754, 21211008, 21337765, 21465026, 21592792, 21721064, 21849843, 21979130, 22108926, 22239232, 22370049, 22501378, 22633220, 22765576, 22898447, 23031834, 23165738, 23300160, 23435101, 23570562, 23706544, 23843048, 23980075, 24117626, 24255702, 24394304, 24533433, 24673090, 24813276, 24953992, 25095239, 25237018, 25379330, 25522176, 25665557, 25809474, 25953928, 26098920, 26244451, 26390522, 26537134, 26684288, 26831985, 26980226, 27129012, 27278344, 27428223, 27578650, 27729626, 27881152, 28033229, 28185858, 28339040, 28492776, 28647067, 28801914, 28957318, 29113280, 29269801, 29426882, 29584524, 29742728, 29901495, 30060826, 30220722, 30381184, 30542213, 30703810, 30865976, 31028712, 31192019, 31355898, 31520350, 31685376, 31850977, 32017154, 32183908, 32351240, 32519151, 32687642, 32856714, 33026368, 33196605, 33367426, 33538832, 33710824, 33883403, 34056570, 34230326, 34404672, 34579609, 34755138, 34931260, 35107976, 35285287, 35463194, 35641698, 35820800.

另A000125给出1000个蛋糕数。

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A003600给出1000个圆环或百吉饼（都有一定的厚度和中心小圆孔的圆环体）切割n刀的最多切块数。
Maximal number of pieces obtained by slicing a torus (or a bagel) with n cuts: (n^3 + 3*n^2 + 8*n)/6 (n > 0).
将圆环（或百吉饼）切 n 次获得的最大切片数：(n^3 + 3*n^2 + 8*n)/6 (n > 0)。

1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, 297, 376, 468, 574, 695, 832, 986, 1158, 1349, 1560, 1792, 2046, 2323, 2624, 2950, 3302, 3681, 4088, 4524, 4990, 5487, 6016, 6578, 7174, 7805, 8472, 9176, 9918, 10699, 11520, 12382, 13286, 14233, 15224 ……

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在这个网络时代，yangchuanju先生就是当代福尔摩斯，现在版得，狄仁杰！

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三刀切13块

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