Comments on A028307 From: Fiorentini Mauro Via Rembrandt 2, 20148 Milano Italy July 03, 2001 I have new data, but no deeper insight into the sequence: multiple triangles with the same max value are now quite common and a sort of exponential growth is apparent (and quite logical also). A formula to generate or even approximate the n-th term is still lacking. Best regards Mauro Fiorentini Further terms: 1 3 8 20 43 98 212 465 1000 2144 4497 9504 19872 41455 85356 178630 363467 757085 1541998 3183600 6515066 13357593 27432649 55914902 114683858 233517515 478061719 972479046 1986013932 4035927124 8220897522 16688919560 33888211971 68872731451 139475317115 283891334541 573540699258 1167078894485 2356442986979 4789098166164 9673531160479 19618700561874 39678885974779 80306572024145 162626770539013 328405697554789 664224912124716 1342289189106478 2711239808935289 5481905622876486 The first rows: Line length = 1, min = 1, number of patterns: 1 1 Line length = 2, min = 3, number of patterns: 1 1 2 Line length = 3, min = 8, number of patterns: 1 2 1 4 Line length = 4, min = 20, number of patterns: 3 4 1 2 7 7 1 2 4 6 1 3 2 Line length = 5, min = 43, number of patterns: 1 7 2 1 4 6 Line length = 6, min = 98, number of patterns: 1 8 6 1 3 2 10 Line length = 7, min = 212, number of patterns: 1 11 7 2 1 4 6 13 Line length = 8, min = 465, number of patterns: 1 16 8 6 1 3 2 10 15 Line length = 9, min = 1000, number of patterns: 2 21 11 7 2 1 4 6 13 17 17 11 7 2 1 4 6 13 21 Line length = 10, min = 2144, number of patterns: 1 23 16 8 6 1 3 2 10 15 18 Line length = 11, min = 4497, number of patterns: 1 20 18 9 8 3 1 5 2 12 19 30 Line length = 12, min = 9504, number of patterns: 1 21 16 15 10 4 1 2 7 6 11 18 23 Line length = 13, min = 19872, number of patterns: 1 26 19 18 9 8 3 1 5 2 12 20 22 16 Line length = 14, min = 41455, number of patterns: 1 24 21 16 15 10 4 1 2 7 6 11 18 23 26 Line length = 15, min = 85356, number of patterns: 1 37 24 22 19 2 10 1 5 3 4 9 16 18 26 28 Line length = 16, min = 178630, number of patterns: 1 31 29 21 20 14 9 4 2 3 7 1 16 12 27 22 24 Line length = 17, min = 363467, number of patterns: 1 30 37 24 22 19 2 10 1 5 3 4 9 16 18 26 28 27 Line length = 18, min = 757085, number of patterns: 2 37 27 26 24 16 14 4 6 3 5 2 11 1 21 23 29 31 38 42 27 26 24 16 14 4 6 3 5 2 11 1 21 23 29 31 33 Line length = 19, min = 1541998, number of patterns: 1 39 30 37 24 22 19 2 10 1 5 3 4 9 16 18 26 28 27 36 Line length = 20, min = 3183600, number of patterns: 1 41 37 27 26 24 16 14 4 6 3 5 2 11 1 21 23 29 31 38 33 Line length = 21, min = 6515066, number of patterns: 1 42 32 36 30 24 22 19 2 10 1 5 3 4 9 16 18 27 26 37 43 39 Line length = 22, min = 13357593, number of patterns: 1 46 41 37 27 26 24 16 14 4 6 3 5 2 11 1 21 23 29 31 38 33 39 Line length = 23, min = 27432649, number of patterns: 1 47 43 32 36 30 24 22 19 2 10 1 5 3 4 9 16 18 27 26 39 28 42 44 Line length = 24, min = 55914902, number of patterns: 1 42 46 41 37 27 26 24 16 14 4 6 3 5 2 11 1 21 23 29 31 38 33 39 43 Line length = 25, min = 114683858, number of patterns: 1 48 44 43 40 29 32 25 17 19 3 8 4 2 5 9 1 15 20 27 28 38 30 37 39 46 Line length = 26, min = 233517515, number of patterns: 2 49 42 46 41 37 27 26 24 16 14 4 6 3 5 2 11 1 21 23 29 31 38 33 39 43 51 51 42 46 41 37 27 26 24 16 14 4 6 3 5 2 11 1 21 23 29 31 38 33 39 43 49 Line length = 27, min = 478061719, number of patterns: 1 53 48 44 43 40 29 32 25 17 19 3 8 4 2 5 9 1 15 20 27 28 38 30 37 39 46 52 Line length = 28, min = 972479046, number of patterns: 2 53 51 46 49 42 34 32 30 20 18 4 10 5 3 6 1 11 2 24 21 27 28 31 41 37 43 47 58 54 51 46 49 42 34 32 30 20 18 4 10 5 3 6 1 11 2 24 21 27 28 31 41 37 43 47 57 Line length = 29, min = 1986013932, number of patterns: 2 59 55 45 47 43 38 32 34 19 22 6 14 3 5 2 9 1 12 4 26 24 27 31 36 35 42 57 49 56 56 55 45 47 43 38 32 34 19 22 6 14 3 5 2 9 1 12 4 26 24 27 31 36 35 42 57 49 59 Line length = 30, min = 4035927124, number of patterns: 1 57 56 46 49 42 43 34 32 30 20 18 4 10 5 3 6 1 11 2 24 21 27 28 31 41 37 53 47 51 54 Line length = 31, min = 8220897522, number of patterns: 2 54 59 55 45 47 43 38 32 34 19 22 6 14 3 5 2 9 1 12 4 26 24 27 31 36 35 42 57 49 56 60 60 56 55 45 47 43 38 32 34 19 22 6 14 3 5 2 9 1 12 4 26 24 27 31 36 35 42 57 49 59 54 Line length = 32, min = 16688919560, number of patterns: 1 57 58 52 50 48 36 38 35 32 30 26 5 18 2 6 3 4 10 1 12 15 19 22 29 39 42 47 44 55 53 59 63 Line length = 33, min = 33888211971, number of patterns: 1 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 Line length = 34, min = 68872731451, number of patterns: 1 60 57 58 52 50 48 36 38 35 32 30 26 5 18 2 6 3 4 10 1 12 15 19 22 29 39 42 47 44 55 53 59 63 65 Line length = 35, min = 139475317115, number of patterns: 1 69 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 66 Line length = 36, min = 283891334541, number of patterns: 1 66 60 57 58 52 50 48 36 38 35 32 30 26 5 18 2 6 3 4 10 1 12 15 19 22 29 39 42 47 44 55 53 59 63 65 69 Line length = 37, min = 573540699258, number of patterns: 2 71 69 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 66 70 70 69 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 66 71 Line length = 38, min = 1167078894485, number of patterns: 1 71 69 63 61 59 53 60 42 49 37 36 34 22 9 24 1 13 2 4 7 5 3 16 10 18 30 32 35 44 41 52 46 55 51 57 66 65 74 Line length = 39, min = 2356442986979, number of patterns: 1 73 71 69 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 66 70 72 Line length = 40, min = 4789098166164, number of patterns: 1 79 74 70 69 62 60 54 49 50 45 38 37 28 23 18 14 7 6 5 3 9 1 15 2 25 4 31 30 36 40 46 47 55 57 52 63 67 68 78 71 Line length = 41, min = 9673531160479, number of patterns: 1 79 73 71 69 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 66 70 72 77 Line length = 42, min = 19618700561874, number of patterns: 2 80 79 74 70 69 62 60 54 49 50 45 38 37 28 23 18 14 7 6 5 3 9 1 15 2 25 4 31 30 36 40 46 47 55 57 52 63 67 68 78 71 85 85 79 74 70 69 62 60 54 49 50 45 38 37 28 23 18 14 7 6 5 3 9 1 15 2 25 4 31 30 36 40 46 47 55 57 52 63 67 68 78 71 80 Line length = 43, min = 39678885974779, number of patterns: 1 86 79 73 71 69 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 66 70 72 77 83 Line length = 44, min = 80306572024145, number of patterns: 2 84 82 77 78 62 68 69 58 52 51 45 43 40 33 37 18 21 4 15 2 11 3 5 7 9 1 23 6 32 31 41 35 46 47 50 59 56 61 65 71 74 75 79 85 85 82 77 78 62 68 69 58 52 51 45 43 40 33 37 18 21 4 15 2 11 3 5 7 9 1 23 6 32 31 41 35 46 47 50 59 56 61 65 71 74 75 79 84 Line length = 45, min = 162626770539013, number of patterns: 1 88 86 79 73 71 69 64 63 52 55 45 48 46 36 32 25 28 6 18 1 13 2 5 4 8 3 17 10 30 26 35 39 41 50 42 60 49 59 65 66 70 72 77 83 84 Line length = 46, min = 328405697554789, number of patterns: 2 86 84 82 77 78 62 68 69 58 52 51 45 43 40 33 37 18 21 4 15 2 11 3 5 7 9 1 23 6 32 31 41 35 46 47 50 59 56 61 65 71 74 75 79 85 92 92 85 82 77 78 62 68 69 58 52 51 45 43 40 33 37 18 21 4 15 2 11 3 5 7 9 1 23 6 32 31 41 35 46 47 50 59 56 61 65 71 74 75 79 84 86 Line length = 47, min = 664224912124716, number of patterns: 1 96 88 82 79 80 71 68 66 61 54 57 50 43 46 39 38 32 3 25 5 10 6 7 4 8 1 17 2 20 14 26 36 33 42 52 49 55 51 59 65 67 73 72 81 84 83 90 Line length = 48, min = 1342289189106478, number of patterns: 2 89 86 84 82 77 78 62 68 69 58 52 51 45 43 40 33 37 18 21 4 15 2 11 3 5 7 9 1 23 6 32 31 41 35 46 47 50 59 56 61 65 71 74 75 79 85 92 99 99 92 85 82 77 78 62 68 69 58 52 51 45 43 40 33 37 18 21 4 15 2 11 3 5 7 9 1 23 6 32 31 41 35 46 47 50 59 56 61 65 71 74 75 79 84 86 89 Line length = 49, min = 2711239808935289, number of patterns: 1 98 96 88 82 79 80 71 68 66 61 54 57 50 43 46 39 38 32 3 25 5 10 6 7 4 8 1 17 2 20 14 26 36 33 42 52 49 55 51 59 65 67 73 72 81 84 83 90 87 Line length = 50, min = 5481905622876486, number of patterns: 1 96 90 88 84 85 79 74 66 67 61 59 58 53 44 43 38 37 34 21 27 2 16 1 10 3 6 8 4 15 5 25 7 33 36 42 41 51 54 56 60 65 64 73 68 76 86 80 93 95 99