Integer valued right triangles and integer, half-integer valued right triangles

Something I figured out as an eight year old child but never published. I’m sure it has been done before, but it is kind of neat.

Integer, half-integer right triangles
A^2 = B+C, where C=B+1, and A is an even integer
Check: is C^2 – B^2 – A^2 = 0 ?
A B C=B+1 C*C B*B A*A Check:
4 7.5 8.5 72.25 56.25 16 0
6 17.5 18.5 342.25 306.25 36 0
8 31.5 32.5 1056.25 992.25 64 0
10 49.5 50.5 2550.25 2450.25 100 0
12 71.5 72.5 5256.25 5112.25 144 0
14 97.5 98.5 9702.25 9506.25 196 0
16 127.5 128.5 16512.25 16256.25 256 0
18 161.5 162.5 26406.25 26082.25 324 0
20 199.5 200.5 40200.25 39800.25 400 0
22 241.5 242.5 58806.25 58322.25 484 0
24 287.5 288.5 83232.25 82656.25 576 0
26 337.5 338.5 114582.3 113906.3 676 0
28 391.5 392.5 154056.3 153272.3 784 0
30 449.5 450.5 202950.3 202050.3 900 0
32 511.5 512.5 262656.3 261632.3 1024 0
34 577.5 578.5 334662.3 333506.3 1156 0
36 647.5 648.5 420552.3 419256.3 1296 0
38 721.5 722.5 522006.3 520562.3 1444 0
40 799.5 800.5 640800.3 639200.3 1600 0
42 881.5 882.5 778806.3 777042.3 1764 0
44 967.5 968.5 937992.3 936056.3 1936 0
46 1057.5 1058.5 1120422 1118306 2116 0
48 1151.5 1152.5 1328256 1325952 2304 0
50 1249.5 1250.5 1563750 1561250 2500 0
52 1351.5 1352.5 1829256 1826552 2704 0
54 1457.5 1458.5 2127222 2124306 2916 0
56 1567.5 1568.5 2460192 2457056 3136 0
58 1681.5 1682.5 2830806 2827442 3364 0
60 1799.5 1800.5 3241800 3238200 3600 0
62 1921.5 1922.5 3696006 3692162 3844 0
64 2047.5 2048.5 4196352 4192256 4096 0  ……….

 

Integer valued right triangles
A^2 = B+C, where C=B+1, where A is an odd integer
Check: is C^2 – B^2 – A^2 = 0 ?
A B C=B+1 C*C B*B A*A Check:
3 4 5 25 16 9 0
5 12 13 169 144 25 0
7 24 25 625 576 49 0
9 40 41 1681 1600 81 0
11 60 61 3721 3600 121 0
13 84 85 7225 7056 169 0
15 112 113 12769 12544 225 0
17 144 145 21025 20736 289 0
19 180 181 32761 32400 361 0
21 220 221 48841 48400 441 0
23 264 265 70225 69696 529 0
25 312 313 97969 97344 625 0
27 364 365 133225 132496 729 0
29 420 421 177241 176400 841 0
31 480 481 231361 230400 961 0
33 544 545 297025 295936 1089 0
35 612 613 375769 374544 1225 0
37 684 685 469225 467856 1369 0
39 760 761 579121 577600 1521 0
41 840 841 707281 705600 1681 0
43 924 925 855625 853776 1849 0
45 1012 1013 1026169 1024144 2025 0
47 1104 1105 1221025 1218816 2209 0
49 1200 1201 1442401 1440000 2401 0
51 1300 1301 1692601 1690000 2601 0
53 1404 1405 1974025 1971216 2809 0
55 1512 1513 2289169 2286144 3025 0
57 1624 1625 2640625 2637376 3249 0
59 1740 1741 3031081 3027600 3481 0
61 1860 1861 3463321 3459600 3721 0
63 1984 1985 3940225 3936256 3969 0 …………….
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