パスカル三角形をご存じであるとしよう。以下のような自然数の生成である。それぞれが二項係数になっている。ｎ段目の前からk番目の二項係数（n,k）だ。
これを虚数の指数に適用する。（n,k）は二項係数でありますな。
上記の三角形に対応する計算結果を示す。当然ながら、１，-１，I、-Iが出廻る。
ここからが高等お遊戯であります。各々の段で総和がどうなるかを追跡してみます。
最初の20段での結果。
I, 2 I, -1 + 2 I, 0, 1 + 2 I, -2 + 4 I, -1, 0, 5 + 2 I, 2 + 4 I, -1 + 4 I, -4, 5, -2, -1, 0, 13 + 2 I, 10 + 4 I, 7 + 4 I, 4, 9 + 4 I
時おり、整数が生まれていますがどういうパターンで出ているのでしょうか？
最初の１００段では６７個の整数があります。
0, -1, 0, -4, 5, -2, -1, 0, 4, -1, -8, 17, 10, 3, -4, 13, -2, -1, 0, 20, 15, 8, -4, 21, -2, -1, -16, 41, 34, 27, 20, 29, 6, 7, -8, 41, 26, 11, -4, 29, -2, -1, 0, 52, 47, 40, 28, 53, 30, 31, 16, 36, 23, -8, 65, 42, 27, -4, 45, -2, -1, -32, 89, 82, 75, 68, 77
負数はあまり大きくならないで正数側が成長している感じですが、それはプロットしてみると裏付けできます。
捜査範囲をもっと大きくします（n=3000）。こうすると何らかの規則性が透けて見えるようです。
ジグザグの降下も上昇もある範囲内で繰り返されるように思えます。
では、どんな自然数が含まれており、どんな自然数がそうでないか？
あるいは、どの段で整数になるのか（こちらのほうが、易しい問題でしょう→計算事例は末尾）？
どんな整数が頻出するのか？
本当はそれらが気になる問題ですね。オープン・クエスチョンであります。
参考までに２０００までの自然数をソートして添付します。
3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 15, 16, 17, 20, 20, 21, 22, 23, 23, 24, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 34, 36, 40, 40, 41, 41, 42, 43, 44, 44, 45, 46, 46, 47, 47, 47, 48, 52, 52, 53, 54, 54, 54, 55, 55, 55, 56, 56, 58, 58, 59, 59, 60, 60, 61,
61, 62, 62, 63, 63, 64, 65, 67, 68, 69, 72, 75, 77, 80, 80, 81, 82,82, 84, 86, 87, 87, 88, 88, 89, 89, 90, 91, 91, 92, 92, 92, 93, 94,94, 94, 95, 95, 95, 96, 97, 100, 103, 104, 104, 106, 107, 108, 108,108, 109, 109, 110, 110, 110, 111, 111, 111, 111, 112, 112, 115, 116,
116, 116, 116, 117, 117, 118, 118, 119, 119, 120, 120, 122, 122, 123,123, 124, 124, 125, 125, 126, 126, 127, 127, 128, 129, 130, 134, 135,136, 137, 138, 139, 141, 144, 147, 147, 149, 149, 150, 151, 153, 154,157, 160, 160, 162, 164, 164, 168, 169, 171, 172, 172, 173, 174, 174,175, 175, 176, 176, 177, 178, 178, 179, 180, 181, 182, 182, 183, 183,183, 184, 184, 184, 185, 185, 185, 185, 186, 187, 187, 187, 188, 188,188, 189, 190, 190, 190, 191, 191, 191, 192, 194, 200, 203, 204, 206,207, 208, 208, 209, 209, 212, 212, 214, 215, 215, 216, 216, 216, 218,218, 219, 219, 220, 220, 220, 220, 221, 221, 222, 222, 222, 222, 223,
223, 223, 223, 224, 224, 228, 230, 231, 231, 232, 232, 232, 232, 234,234, 235, 236, 236, 237, 237, 237, 238, 238, 239, 239, 239, 240, 240,243, 243, 244, 244, 244, 244, 245, 245, 245, 246, 246, 247, 247, 248,248, 250, 250, 251, 251, 252, 252, 253, 253, 254, 254, 255, 255, 256,257, 258, 260, 263, 265, 267, 268, 269, 270, 271, 272, 274, 276, 276,278, 279, 281, 281, 281, 282, 283, 285, 288, 291, 293, 294, 294, 295,295, 297, 298, 298, 300, 301, 301, 302, 303, 306, 307, 307, 308, 309,309, 314, 317, 320, 320, 321, 324, 328, 328, 329, 331, 331, 336, 338, 339, 339, 341, 341, 342, 343, 344, 344, 345, 346, 348, 348, 349, 350,
350, 351, 351, 352, 352, 354, 355, 356, 356, 357, 358, 359, 360, 361, 361, 362, 363, 364, 364, 364, 365, 365, 366, 366, 366, 367, 367, 367, 368, 368, 368, 369, 370, 370, 370, 370, 372, 374, 374, 374, 375, 375, 375, 376, 376, 376, 377, 377, 377, 377, 378, 379, 379, 379, 380, 380, 380, 381, 382, 382, 382, 383, 383, 383, 385, 387, 388, 389, 393, 393, 400, 403, 405, 406, 407, 408, 409, 411, 412, 414, 415, 416, 416, 417, 418, 418, 424, 424, 427, 428, 428, 430, 430, 431, 431, 432, 432, 432, 433, 433, 436, 436, 438, 438, 439, 439, 439, 440, 440, 440, 440, 442, 442, 443, 443, 443, 444, 444, 444, 444, 445, 445, 446, 446, 446, 446,
447, 447, 447, 447, 448, 449, 456, 459, 460, 460, 462, 462, 463, 463, 464, 464, 464, 464, 468, 468, 468, 470, 471, 471, 472, 474, 474, 474, 475, 475, 476, 476, 477, 477, 477, 478, 478, 479, 479, 480, 484, 486, 486, 487, 487, 487, 488, 488, 488, 488, 490, 490, 490, 491, 492, 493, 493, 493, 493, 494, 495, 495, 496, 499, 499, 499, 500, 500, 500, 500, 501, 501, 501, 501, 502, 503, 503, 504, 506, 506, 507, 507, 508, 509, 509, 510, 511, 513, 514, 516, 519, 519, 520, 521, 524, 525, 525, 526, 527, 530, 531, 532, 532, 533, 534, 535, 536, 537, 537, 537, 538, 539, 540, 541, 542, 543, 544, 547, 548, 548, 549, 552, 552, 553, 555, 555, 556, 558, 559, 562, 562, 562, 563, 564, 565, 566, 567, 569, 569, 570, 571, 573, 576, 577, 582, 583, 583, 585, 586, 588, 588, 589, 589, 590, 590, 591, 591, 594, 596, 596, 596, 600, 601, 601, 602, 602, 604, 605, 605, 606, 607, 611, 612, 613, 614, 614, 615, 615, 616, 617, 618, 618, 619, 621, 621, 625, 627, 627, 628, 629, 629, 634, 637, 640, 640, 642, 643, 645, 648, 649, 651, 656, 656, 657, 657, 658, 662, 662, 663, 663, 667, 667, 672, 676, 678, 678, 679, 679, 681, 682, 682, 683, 684, 685, 685, 686, 687, 688, 688, 689, 690, 691, 691, 691, 692, 693, 693, 693, 696, 696, 697, 698, 700, 700, 701, 702, 702, 703, 703, 708, 710, 711,
712, 712, 714, 715, 715, 716, 717, 718, 719, 720, 722, 722, 723, 723, 724, 725, 725, 726, 727, 728, 728, 728, 729, 729, 730, 730, 732, 732, 732, 733, 733, 734, 734, 734, 735, 735, 735, 736, 738, 739, 739, 740, 740, 740, 740, 741, 741, 744, 745, 745, 745, 747, 748, 748, 748, 749, 750, 750, 750, 751, 751, 751, 752, 753, 754, 754, 754, 754, 756, 758, 758, 758, 759, 759, 759, 760, 761, 761, 761, 761, 762, 763, 763, 763, 764, 765, 766, 767, 769, 770, 771, 773, 774, 775, 776, 777, 777, 778, 779, 781, 785, 785, 786, 786, 787, 789, 793, 800, 803, 803, 805, 805, 806, 807, 809, 809, 809, 810, 812, 813, 814, 815, 816, 817, 818, 822,
823, 824, 825, 827, 828, 830, 831, 832, 832, 834, 836, 836, 848, 848, 849, 851, 853, 854, 855, 856, 856, 857, 859, 860, 860, 862, 862, 863, 863, 864, 865, 866, 866, 867, 869, 872, 872, 873, 875, 876, 876, 876, 878, 878, 878, 879, 879, 879, 880, 880, 881, 881, 881, 881, 884, 884, 886, 886, 886, 887, 887, 887, 888, 888, 890, 890, 891, 891, 891, 892, 892, 893, 893, 894, 894, 895, 895, 898, 912, 918, 919, 920, 920, 923, 924, 924, 926, 926, 927, 927, 928, 928, 928, 928, 929, 929, 936, 936, 936, 939, 940, 940, 942, 942, 943, 943, 948, 948, 948, 950, 950, 951, 951, 951, 952, 954, 954, 954, 955, 955, 955, 956, 957, 957, 957, 958, 959, 968, 971, 972, 972, 972, 974, 974, 974, 975, 975, 975, 976, 976, 976, 976, 980, 980, 980, 980, 983, 983, 986, 986, 986, 986, 987, 987, 988, 989, 989, 989, 989, 990, 991, 996, 998, 998, 998, 999, 999, 999, 999, 1000, 1000, 1000, 1002, 1002, 1002, 1002, 1003, 1004, 1005, 1005, 1005, 1005, 1005, 1006, 1007, 1007, 1011, 1011, 1011, 1011, 1012, 1012, 1012, 1013, 1013, 1013, 1013, 1013, 1014, 1015, 1015, 1018, 1019, 1019, 1021, 1025, 1026, 1028, 1031, 1031, 1032, 1033, 1036, 1036, 1037, 1037, 1038, 1038, 1039, 1039, 1040, 1042, 1043, 1044, 1044, 1044, 1045, 1048, 1049, 1049, 1049, 1050, 1050, 1052, 1053, 1053, 1054, 1055, 1059, 1060, 1060, 1061, 1062, 1063, 1064, 1064, 1065, 1066, 1067, 1067, 1068, 1069, 1070, 1071, 1072, 1074, 1074, 1074, 1075, 1075, 1076, 1077, 1077, 1078, 1079, 1080, 1081, 1081, 1082, 1083, 1084, 1085, 1086, 1087, 1088, 1089, 1094, 1095, 1095, 1096, 1096, 1097, 1098, 1101, 1101, 1104, 1104, 1106, 1108, 1110, 1110, 1111, 1111, 1112, 1113, 1113, 1115, 1115, 1116, 1118, 1119, 1123, 1124, 1124, 1124, 1125, 1126, 1127, 1128, 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1137, 1138, 1138, 1139, 1140, 1141, 1142, 1143, 1143, 1145, 1145, 1146, 1147, 1147, 1149, 1154, 1155, 1157, 1159, 1161, 1163, 1164, 1164, 1165, 1166, 1166, 1167, 1167, 1169, 1169, 1170, 1171, 1171, 1172, 1172, 1173, 1173, 1176, 1176, 1178, 1178, 1180, 1180, 1181, 1181, 1182, 1182, 1183, 1183, 1188, 1188, 1191, 1192, 1192, 1192, 1193, 1197, 1200, 1201, 1202, 1202, 1204, 1204, 1208, 1209, 1209, 1209, 1210, 1210, 1212, 1213, 1213,
1214, 1215, 1219, 1220, 1221, 1222, 1223, 1223, 1224, 1225, 1226, 1228, 1228, 1229, 1229, 1230, 1230, 1231, 1231, 1232, 1234, 1236, 1236, 1238, 1239, 1241, 1241, 1242, 1243, 1245, 1250, 1251, 1253, 1254, 1255, 1256, 1257, 1258, 1259, 1259, 1261, 1265, 1265, 1267, 1267, 1268, 1269, 1269, 1274, 1277, 1283, 1284, 1285, 1286, 1287, 1289, 1290, 1293, 1296, 1298, 1302, 1303, 1305, 1307, 1312, 1312, 1313, 1314, 1314, 1316, 1323, 1323, 1323, 1324, 1324, 1326, 1326, 1327, 1327, 1329, 1329, 1334, 1334, 1335, 1335, 1339, 1339, 1347, 1347, 1349, 1349, 1352, 1353, 1353, 1356, 1356, 1358, 1358, 1359, 1359, 1361, 1361, 1362, 1364, 1364, 1366, 1367, 1368, 1370, 1370, 1371, 1372, 1373, 1373, 1374, 1375, 1377, 1378, 1380, 1382, 1382, 1382, 1383, 1383, 1383, 1384, 1385, 1386, 1386, 1386, 1389, 1389, 1389, 1394, 1395, 1395, 1395, 1396, 1397, 1397, 1397, 1401, 1402, 1405, 1409, 1416, 1419, 1419, 1420, 1422, 1423, 1424, 1424, 1425, 1427, 1428, 1429, 1430, 1430, 1431, 1431, 1432, 1434, 1435, 1435,
1436, 1437, 1438, 1439, 1444, 1444, 1446, 1446, 1447, 1447, 1448, 1450, 1450, 1451, 1452, 1453, 1453, 1454, 1455, 1458, 1458, 1459, 1459, 1459, 1460, 1460, 1461, 1461, 1461, 1465, 1465, 1466, 1466, 1469, 1469, 1476, 1478, 1478, 1479, 1479, 1480, 1480, 1480, 1480, 1482, 1482, 1483, 1483, 1485, 1485, 1490, 1490, 1490, 1491, 1493, 1494, 1495, 1497, 1497, 1497, 1498, 1501, 1506, 1507, 1507, 1507, 1508, 1508, 1508, 1509, 1509, 1509, 1513, 1513, 1513, 1513, 1515, 1515, 1515, 1517, 1521, 1522, 1522, 1522, 1523, 1525, 1529, 1529, 1529, 1537, 1538, 1539, 1539, 1540, 1541, 1541, 1542, 1543, 1545, 1545, 1545, 1546, 1547, 1548, 1549, 1550, 1551, 1552, 1553, 1553, 1554, 1554, 1555, 1556, 1557, 1558, 1559, 1561, 1562, 1563, 1565, 1570, 1570, 1572, 1572, 1574, 1575, 1577, 1578, 1581, 1585, 1585, 1585, 1586, 1587, 1589, 1593, 1603, 1603, 1605, 1605, 1606, 1606, 1607, 1607, 1609, 1609, 1610, 1610, 1612, 1613, 1613, 1614, 1615, 1617, 1617, 1618, 1618, 1620, 1625, 1626, 1629, 1633, 1634, 1635,
1636, 1637, 1641, 1641, 1650, 1657, 1665, 1668, 1672, 1672, 1675, 1675, 1681, 1683, 1683, 1685, 1685, 1698, 1699, 1699, 1701, 1701, 1702, 1703, 1705, 1705, 1706, 1709, 1713, 1714, 1721, 1730, 1731, 1732, 1732, 1733, 1734, 1735, 1737, 1738, 1741, 1745, 1745, 1746, 1747, 1749, 1753, 1753, 1761, 1762, 1762, 1762, 1777, 1777, 1777, 1793, 1795, 1796, 1797, 1801, 1809, 1809, 1827, 1829, 1833, 1841, 1857, 1858, 1858, 1889, 1921
総和が整数になるｎ 最初の１００まで
3, 6, 7, 11, 12, 13, 14, 15, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 35, 38, 39, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 70, 71, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
