Four Fun Facts About Triangular Numbers

In my previous post (Happy Triangular Numbers) I made reference to a website Fascinating Triangular Numbers and this post I'd like to mention just four more of the "fun facts" mentioned there.

FUN FACT 1: 

The sum of two consecutive triangular numbers is a square number. This is easily proven as follow:

$$\begin{align} T_n+T_{n+1} &= \frac{n(n+1)}{2}+ \frac{(n+1)(n+2)}{2}\\ &= \frac{n+1}{2} \cdot (2n+2)\\ &= (n+1)^2 \end{align}$$ FUN FACT 2:

The sum of the squares of two consecutive triangular numbers is also a triangular number. Again this is easily proven as follows:

$$\begin{align} \big (T_n \big )^2+ \big (T_{n+1} \big )^2&= \big (\frac{n(n+1)}{2} \big )^2+ \big ( \frac{(n+1)(n+2)}{2} \big )^2\\ &= \Big (\frac{n+1}{2} \Big )^2 \cdot \Big ( n^2+(n+2)^2 \Big )\\ &= \Big (\frac{n+1}{2} \Big )^2 \cdot \Big ( 2n^2+4n+4 \Big ) \\ &= \frac {(n^2 +2n+1) \cdot (n^2+2n+2)}{2} \\ &=T_{(n+1)^2} \end{align}$$ FUN FACT 3:

There are infinitely many triangular numbers, which are also squares as given by the series 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056 etc. These can be termed as square triangular numbers. The $n$th Square Triangular number $K_n$ can easily be obtained from the recursive formula: 

$$K_n = 34 \times K_{n-1} - K_{n-2} + 2$$ So knowing the first two square triangular numbers i.e. $K_1 = 1$ and $K_2 = 36$ , all other successive Square Triangular numbers can be obtained. For example: $$\begin{align} K_3 &= 34 \times K_2 - K_1 + 2 \\ &= 34 \times 36 -1 + 2 \\ &= 1225 \\ &= 35^2 \\ K_4 &= 34 \times K_3 - K_2 + 2 \\ &= 34 \times 1225 - 36 + 2 \\ &= 41616 \\ &=204^2 \end{align}$$ FUN FACT 4:

There exist infinite triangular numbers that are simultaneously the sum, the difference and the product of two other triangular numbers. Here is a list of the initial such numbers:

$$\begin {align} 990 &= 1035 - 45 = 780 + 210 = 66 \times 15\\ 1540 &= 1711 - 171 = 1485 + 55 = 55 \times 28\\ 2850 &= 3003 - 153 = 2415 + 435 = 190 \times 15\\ 4851 &= 5151 - 300 = 3081 + 1770 = 231 \times 21\\ 8778 &= 10731 - 1953 = 7875 + 903 = 2926 \times 3\\ 11781 &= 12246 - 465 = 11628 +153 = 561 \times 21\\ 15400 &= 18721 - 3321 = 14365 + 1035 =1540 \times 10\\ 26796 &= 27261 - 465 = 26565 + 231 = 406 \times 66\\ 43956 &= 44551 - 595 = 41328 + 2628 = 666 \times 66 \end{align}$$ Here are the initial triangular numbers along with their associated indices (permalink):

[(1, 1), (3, 2), (6, 3), (10, 4), (15, 5), (21, 6), (28, 7), (36, 8), (45, 9), (55, 10), (66, 11), (78, 12), (91, 13), (105, 14), (120, 15), (136, 16), (153, 17), (171, 18), (190, 19), (210, 20), (231, 21), (253, 22), (276, 23), (300, 24), (325, 25), (351, 26), (378, 27), (406, 28), (435, 29), (465, 30), (496, 31), (528, 32), (561, 33), (595, 34), (630, 35), (666, 36), (703, 37), (741, 38), (780, 39), (820, 40), (861, 41), (903, 42), (946, 43), (990, 44), (1035, 45), (1081, 46), (1128, 47), (1176, 48), (1225, 49), (1275, 50), (1326, 51), (1378, 52), (1431, 53), (1485, 54), (1540, 55), (1596, 56), (1653, 57), (1711, 58), (1770, 59), (1830, 60), (1891, 61), (1953, 62), (2016, 63), (2080, 64), (2145, 65), (2211, 66), (2278, 67), (2346, 68), (2415, 69), (2485, 70), (2556, 71), (2628, 72), (2701, 73), (2775, 74), (2850, 75), (2926, 76), (3003, 77), (3081, 78), (3160, 79), (3240, 80), (3321, 81), (3403, 82), (3486, 83), (3570, 84), (3655, 85), (3741, 86), (3828, 87), (3916, 88), (4005, 89), (4095, 90), (4186, 91), (4278, 92), (4371, 93), (4465, 94), (4560, 95), (4656, 96), (4753, 97), (4851, 98), (4950, 99), (5050, 100), (5151, 101), (5253, 102), (5356, 103), (5460, 104), (5565, 105), (5671, 106), (5778, 107), (5886, 108), (5995, 109), (6105, 110), (6216, 111), (6328, 112), (6441, 113), (6555, 114), (6670, 115), (6786, 116), (6903, 117), (7021, 118), (7140, 119), (7260, 120), (7381, 121), (7503, 122), (7626, 123), (7750, 124), (7875, 125), (8001, 126), (8128, 127), (8256, 128), (8385, 129), (8515, 130), (8646, 131), (8778, 132), (8911, 133), (9045, 134), (9180, 135), (9316, 136), (9453, 137), (9591, 138), (9730, 139), (9870, 140), (10011, 141), (10153, 142), (10296, 143), (10440, 144), (10585, 145), (10731, 146), (10878, 147), (11026, 148), (11175, 149), (11325, 150), (11476, 151), (11628, 152), (11781, 153), (11935, 154), (12090, 155), (12246, 156), (12403, 157), (12561, 158), (12720, 159), (12880, 160), (13041, 161), (13203, 162), (13366, 163), (13530, 164), (13695, 165), (13861, 166), (14028, 167), (14196, 168), (14365, 169), (14535, 170), (14706, 171), (14878, 172), (15051, 173), (15225, 174), (15400, 175), (15576, 176), (15753, 177), (15931, 178), (16110, 179), (16290, 180), (16471, 181), (16653, 182), (16836, 183), (17020, 184), (17205, 185), (17391, 186), (17578, 187), (17766, 188), (17955, 189), (18145, 190), (18336, 191), (18528, 192), (18721, 193), (18915, 194), (19110, 195), (19306, 196), (19503, 197), (19701, 198), (19900, 199), (20100, 200), (20301, 201), (20503, 202), (20706, 203), (20910, 204), (21115, 205), (21321, 206), (21528, 207), (21736, 208), (21945, 209), (22155, 210), (22366, 211), (22578, 212), (22791, 213), (23005, 214), (23220, 215), (23436, 216), (23653, 217), (23871, 218), (24090, 219), (24310, 220), (24531, 221), (24753, 222), (24976, 223), (25200, 224), (25425, 225), (25651, 226), (25878, 227), (26106, 228), (26335, 229), (26565, 230), (26796, 231), (27028, 232), (27261, 233), (27495, 234), (27730, 235), (27966, 236), (28203, 237), (28441, 238), (28680, 239), (28920, 240), (29161, 241), (29403, 242), (29646, 243), (29890, 244), (30135, 245), (30381, 246), (30628, 247), (30876, 248), (31125, 249), (31375, 250), (31626, 251), (31878, 252), (32131, 253), (32385, 254), (32640, 255), (32896, 256), (33153, 257), (33411, 258), (33670, 259), (33930, 260), (34191, 261), (34453, 262), (34716, 263), (34980, 264), (35245, 265), (35511, 266), (35778, 267), (36046, 268), (36315, 269), (36585, 270), (36856, 271), (37128, 272), (37401, 273), (37675, 274), (37950, 275), (38226, 276), (38503, 277), (38781, 278), (39060, 279), (39340, 280), (39621, 281), (39903, 282), (40186, 283), (40470, 284), (40755, 285), (41041, 286), (41328, 287), (41616, 288), (41905, 289), (42195, 290), (42486, 291), (42778, 292), (43071, 293), (43365, 294), (43660, 295), (43956, 296), (44253, 297), (44551, 298), (44850, 299), (45150, 300)]

The results for FUN FACT 4 could be written in terms of these indices. For example:

$$\begin {align} 990 &= 1035 - 45 = 780 + 210 = 66 \times 15\\ \text{T}_{44} &= \text{T}_{45} -\text{T}_{9} = \text{T}_{39} + \text{T}_{20} = \text{T}_{11} \times \text{T}_{5} \end{align}$$