三角形数的三角形图

王守恩

A062708——有个三角形数的三角形图(网页上找不到类似的图)——你来画得比它大一些(我不会画)——我来出道题——蛮好玩的！谢谢！

题目是这样。记"0"关于"n"的对称数为"a(n)"。譬如
a(1)=11,
a(2)=13,
a(3)=15,
a(4)=17,
a(5)=19,
a(6)=21,
a(7)=23,
a(8)=25,
a(9)=27,
a(10)=56,
a(11)=58,
a(12)=60,
a(13)=62,
a(14)=64,
a(15)=66,
a(16)=68,
a(17)=70,
a(18)=72,
a(19)=74,
a(20)=76,
a(21)=78,
a(22)=80,
a(23)=82,
a(24)=84,
a(25)=86,
a(26)=88,
a(27)=90,
a(28)=137,
a(29)=139,
a(30)=141,
a(31)=143,
a(32)=145,
......
得到一串数——11, 13, 15, 17, 19, 21, 23, 25, 27, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 137, 139, 141, 143, 145,......,求通项公式。

王守恩

得到一串数——11, 13, 15, 17, 19, 21, 23, 25, 27, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 137, 139, 141, 143, 145,......,求通项公式。——这通项公式真还不好找(没头绪)。求助各位。

我们先往前走。记"0"关于"n"的对称数为"a(n)"。如果把 "0", "n", "a(n)" 这条直线拉出来。譬如
b(1)={0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, 3683}
b(2)={0, 2, 13, 33, 62, 100, 147, 203, 268, 342, 425, 517, 618, 728, 847, 975, 1112, 1258, 1413, 1577, 1750, 1932, 2123, 2323, 2532, 2750, 2977, 3213, 3458, 3712}
b(3)={0, 3, 15, 36, 66, 105, 153, 210, 276, 351, 435, 528, 630, 741, 861, 990, 1128, 1275, 1431, 1596, 1770, 1953, 2145, 2346, 2556, 2775, 3003, 3240, 3486, 3741}
b(4)={0, 4, 17, 39, 70, 110, 159, 217, 284, 360, 445, 539, 642, 754, 875, 1005, 1144, 1292, 1449, 1615, 1790, 1974, 2167, 2369, 2580, 2800, 3029, 3267, 3514, 3770}
b(5)={0, 5, 19, 42, 74, 115, 165, 224, 292, 369, 455, 550, 654, 767, 889, 1020, 1160, 1309, 1467, 1634, 1810, 1995, 2189, 2392, 2604, 2825, 3055, 3294, 3542, 3799}
b(6)={0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828}
b(7)={0, 7, 23, 48, 82, 125, 177, 238, 308, 387, 475, 572, 678, 793, 917, 1050, 1192, 1343, 1503, 1672, 1850, 2037, 2233, 2438, 2652, 2875, 3107, 3348, 3598, 3857}
b(8)={0, 8, 25, 51, 86, 130, 183, 245, 316, 396, 485, 583, 690, 806, 931, 1065, 1208, 1360, 1521, 1691, 1870, 2058, 2255, 2461, 2676, 2900, 3133, 3375, 3626, 3886}
b(9)={0, 9, 27, 54, 90, 135, 189, 252, 324, 405, 495, 594, 702, 819, 945, 1080, 1224, 1377, 1539, 1710, 1890, 2079, 2277, 2484, 2700, 2925, 3159, 3402, 3654, 3915}
b(10)={0, 10, 56, 138, 256, 410, 600, 826, 1088, 1386, 1720, 2090, 2496, 2938, 3416, 3930, 4480, 5066, 5688, 6346, 7040, 7770, 8536, 9338, 10176, 11050, 11960, 12906, 13888, 14906}
b(11)={0, 11, 58, 141, 260, 415, 606, 833, 1096, 1395, 1730, 2101, 2508, 2951, 3430, 3945, 4496, 5083, 5706, 6365, 7060, 7791, 8558, 9361, 10200, 11075, 11986, 12933, 13916, 14935}
b(12)={0, 12, 60, 144, 264, 420, 612, 840, 1104, 1404, 1740, 2112, 2520, 2964, 3444, 3960, 4512, 5100, 5724, 6384, 7080, 7812, 8580, 9384, 10224, 11100, 12012, 12960, 13944, 14964}
b(13)={0, 13, 62, 147, 268, 425, 618, 847, 1112, 1413, 1750, 2123, 2532, 2977, 3458, 3975, 4528, 5117, 5742, 6403, 7100, 7833, 8602, 9407, 10248, 11125, 12038, 12987, 13972, 14993}
b(14)={0, 14, 64, 150, 272, 430, 624, 854, 1120, 1422, 1760, 2134, 2544, 2990, 3472, 3990, 4544, 5134, 5760, 6422, 7120, 7854, 8624, 9430, 10272, 11150, 12064, 13014, 14000, 15022}
......
b(23)={0, 23, 82, 177, 308, 475, 678, 917, 1192, 1503, 1850, 2233, 2652, 3107, 3598, 4125, 4688, 5287, 5922, 6593, 7300, 8043, 8822, 9637, 10488, 11375, 12298, 13257, 14252, 15283}
b(24)={0, 24, 84, 180, 312, 480, 684, 924, 1200, 1512, 1860, 2244, 2664, 3120, 3612, 4140, 4704, 5304, 5940, 6612, 7320, 8064, 8844, 9660, 10512, 11400, 12324, 13284, 14280, 15312}
b(25)={0, 25, 86, 183, 316, 485, 690, 931, 1208, 1521, 1870, 2255, 2676, 3133, 3626, 4155, 4720, 5321, 5958, 6631, 7340, 8085, 8866, 9683, 10536, 11425, 12350, 13311, 14308, 15341}
b(26)={0, 26, 88, 186, 320, 490, 696, 938, 1216, 1530, 1880, 2266, 2688, 3146, 3640, 4170, 4736, 5338, 5976, 6650, 7360, 8106, 8888, 9706, 10560, 11450, 12376, 13338, 14336, 15370}
b(27)={0, 27, 90, 189, 324, 495, 702, 945, 1224, 1539, 1890, 2277, 2700, 3159, 3654, 4185, 4752, 5355, 5994, 6669, 7380, 8127, 8910, 9729, 10584, 11475, 12402, 13365, 14364, 15399}
b(28)={0, 28, 137, 327, 598, 950, 1383, 1897, 2492, 3168, 3925, 4763, 5682, 6682, 7763, 8925, 10168, 11492, 12897, 14383, 15950, 17598, 19327, 21137, 23028, 25000, 27053, 29187, 31402, 33698}
b(29)={0, 29, 139, 330, 602, 955, 1389, 1904, 2500, 3177, 3935, 4774, 5694, 6695, 7777, 8940, 10184, 11509, 12915, 14402, 15970, 17619, 19349, 21160, 23052, 25025, 27079, 29214, 31430, 33727}
b(30)={0, 30, 141, 333, 606, 960, 1395, 1911, 2508, 3186, 3945, 4785, 5706, 6708, 7791, 8955, 10200, 11526, 12933, 14421, 15990, 17640, 19371, 21183, 23076, 25050, 27105, 29241, 31458, 33756}
b(31)={0, 31, 143, 336, 610, 965, 1401, 1918, 2516, 3195, 3955, 4796, 5718, 6721, 7805, 8970, 10216, 11543, 12951, 14440, 16010, 17661, 19393, 21206, 23100, 25075, 27131, 29268, 31486, 33785}
b(32)={0, 32, 145, 339, 614, 970, 1407, 1925, 2524, 3204, 3965, 4807, 5730, 6734, 7819, 8985, 10232, 11560, 12969, 14459, 16030, 17682, 19415, 21229, 23124, 25100, 27157, 29295, 31514, 33814}
......
这些数字串却可以有一个共同的通项公式。b(n)=3*b(n-1)-3b(n-2)+b(n-3)。——太诱人！