Triangular numbers between consecutive square numbers

 
 
Let   S_{m} \: (n)   be the n-th m-gonal number,   m-gonal number with index   n

S_3 (n) \; = \; n \,(n + 1)/2   ……….   S_4 (n) \; = \; n^2

S_3 \: (n+1) \; = \; S_3 \: (n) \; + \; (n + 1)
S_4 \: (n+1) \; = \; S_4 \: (n) \; + \; (2 \,n + 1)

 
 
Checking up to 50 the following proposition:

If there are exactly two triangular numbers between   S_4 \: (a+1)   and   S_4 \: (a+2),
  where   a > 0,
there is exactly one triangular number between   S_4 \: (a)   and   S_4 \: (a+1),   and exactly one
triangular number between   S_3 \: (a+2)   and   S_4 \: (a+3)

 
1^2 = 1
…………    T_2 = 3
2^2 = 4
…………    T_3 = 6
3^2 = 9
…………    T_4 = 10
…………    T_5 = 15
4^2 = 16
…………    T_6 = 21
5^2 = 25
…………    T_7 = 28

6^2 = 36 = T_8

…………    T_9 = 45
7^2 = 49
…………    T_{10} = 55
8^2 = 64
…………    T_{11} = 66
…………    T_{12} = 78
9^2 = 81
…………    T_{13} = 91
10^2 = 100
…………    T_{14} = 105
…………    T_{15} = 120
11^2 = 121
…………    T_{16} = 136
12^2 = 144
…………    T_{17} = 153
13^2 = 169
…………    T_{18} = 171
…………    T_{19} = 190
14^2 = 196
…………    T_{20} = 210

15^2 = 225
…………    T_{21} = 231
…………    T_{22} = 253
16^2 = 256
…………    T_{23} = 276
17^2 = 289
…………    T_{24} = 300
18^2 = 324
…………    T_{25} = 325
…………    T_{26} = 351
19^2 = 361
…………    T_{27} = 378
20^2 = 400
…………    T_{28} = 406
…………    T_{29} = 435
21^2 = 441
…………    T_{30} = 465
22^2 = 484
…………    T_{31} = 496
…………    T_{32} = 528
23^2 = 529
…………    T_{33} = 561
24^2 = 576
…………    T_{34} = 595
25^2 = 625
…………    T_{35} = 630
…………    T_{36} = 666
26^2 = 676
…………    T_{37} = 703
27^2 = 729
…………    T_{38} = 741
…………    T_{39} = 780
28^2 = 784
…………    T_{40} = 820
29^2 = 841
…………    T_{41} = 861
30^2 = 900
…………    T_{42} = 903
…………    T_{43} = 946
31^2 = 961
…………    T_{44} = 990
32^2 = 1024
…………    T_{45} = 1035
…………    T_{46} = 1081
33^2 = 1089
…………    T_{47} = 1128
34^2 = 1156
…………    T_{48} = 1176

35^2 = 1225 = T_{49}

…………    T_{50} = 1275
36^2 = 1296
…………    T_{51} = 1326
37^2 = 1369
…………    T_{52} = 1378
…………    T_{53} = 1431
38^2 = 1444
…………    T_{54} = 1485
39^2 = 1521
…………    T_{55} = 1540
…………    T_{56} = 1596
40^2 = 1600
…………    T_{57} = 1653
41^2 = 1681
…………    T_{58} = 1711
42^2 = 1764
…………    T_{59} = 1770
43^2 = 1849
…………    T_{60} = 1830
…………    T_{61} = 1891
44^2 = 1936
…………    T_{62} = 1953
…………    T_{63} = 2016
45^2 = 2025
…………    T_{64} = 2080
46^2 = 2116
…………    T_{65} = 2145
47^2 = 2209
…………    T_{66} = 2211
…………    T_{67} = 2278
48^2 = 2304
…………    T_{68} = 2346
49^2 = 2401
…………    T_{69} = 2415
…………    T_{70} = 2485
50^2 = 2500
…………    T_{71} = 2556
51^2 = 2601

 

Does this pattern continue?
Explain.
 
 
 

 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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