A102698
Number of equilateral triangles with coordinates (x,y,z) in the set {0, 1,...,n}.
Original entry on oeis.org
8, 80, 368, 1264, 3448, 7792, 16176, 30696, 54216, 90104, 143576, 220328, 326680, 471232, 664648, 916344, 1241856, 1655208, 2172584, 2812664, 3598664, 4553800, 5702776, 7075264, 8705088, 10628928, 12880056, 15496616, 18523472, 22003808
Offset: 1
a(1) = 8 because in the unit cube, equilateral triangles are formed by cutting off any one of the 8 corners.
a(2) = 80 because there are 8 unit cubes with 8 each, 8 larger triangles (analogous to the 8 in the unit cube, but twice as big) and also 8 triangles of side length sqrt(6).
- Eugen J. Ionascu and Rodrigo A. Obando, Table of n, a(n) for n = 1..100
- Ray Chandler and Eugen J. Ionascu, A characterization of all equilateral triangles in Z^3, arXiv:0710.0708 [math.NT], 2007.
- Eugen J. Ionascu, Maple program
- Eugen J. Ionascu, A parametrization of equilateral triangles having integer coordinates, J. Integer Seqs., Vol. 10 (2007), #07.6.7.
- Eugen J. Ionascu, Counting all equilateral triangles in {0,1,...,n}^3, Acta Mathematica Universitatis Comenianae, Vol. LXXVII, 1 (2008) p. 129-140.
- Rodrigo A. Obando, Mathematica program
- Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).
A103426
(1/4)*Number of non-degenerate triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.
Original entry on oeis.org
14, 719, 10322, 78973, 412666, 1662616, 5550432, 16056600, 41504082, 97957235, 214501838, 441056849, 859632934, 1599921616, 2860527328, 4937138832, 8259305646, 13437461703, 21322651346, 33080660021, 50283889886, 75023188336
Offset: 1
A103428
(1/12)*Number of non-degenerate obtuse triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.
Original entry on oeis.org
0, 62, 1270, 11266, 63322, 266748, 915720, 2701073, 7077080, 16876415, 37242500, 77038188, 150862354, 281877711, 505585682, 874900010, 1466826558, 2390947859, 3799984292, 5903574820, 8984255594, 13418520513, 19700297034, 28470461533
Offset: 1
Cf.
A190020 (analogous 2-dimensional problem).
A103429
(1/4)*number of acute triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.
Original entry on oeis.org
2, 194, 3434, 29356, 162190, 679654, 2323878, 6839595, 17909922, 42675551, 94125356, 194693240, 381214450, 712191373, 1277323894, 2210486280, 3706015236, 6040816887, 9601083812, 14916225896, 22701123860, 33905935285
Offset: 1
Cf.
A190019 (analogous 2-dimensional problem).
A103427
(1/12) * Number of non-degenerate scalene triangles that can be formed from the points of an (n+1) X (n+1) X (n+1) lattice cube.
Original entry on oeis.org
2, 175, 2904, 23522, 126888, 521475, 1765382, 5153295, 13412318, 31816983, 69951724, 144272314, 281895828, 525712348, 941516596, 1627256650, 2725454906, 4438574843, 7049265930, 10944500376, 16646835858, 24851001712, 36469592898
Offset: 1
A103499
(1/12)*number of right triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.
Original entry on oeis.org
4, 113, 1026, 5273, 20170, 60906, 159798, 371262, 787640, 1550813, 2882994, 5083015, 8610474, 14032370, 22148796, 33984174, 50936912, 74600413, 107204886, 151236555, 209999748, 287230504, 387791652, 516909272, 681578384, 888990683
Offset: 1
A103500
(1/4)*number of non-degenerate isosceles triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.
Original entry on oeis.org
8, 194, 1610, 8407, 32002, 98191, 254286, 596715, 1267128, 2506286, 4646666, 8239907, 13945450, 22784572, 35977540, 55368882, 82940928, 121737174, 174853556, 247158893, 343382312, 470183200, 634503574, 847118119, 1117272006
Offset: 1
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