Comments on A028307
From: Fiorentini Mauro <MFiorentini@etnoteam.it>

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