A103158
(1/2)*number of regular tetrahedra that can be formed using the points in an (n+1) X (n+1) X (n+1) lattice cube.
Original entry on oeis.org
1, 9, 36, 104, 257, 549, 1058, 1896, 3199, 5145, 7926, 11768, 16967, 23859, 32846, 44378, 58977, 77215, 99684, 126994, 159963, 199443, 246304, 301702, 366729, 442587, 530508, 631820, 748121, 880941, 1031930, 1202984, 1395927, 1612655, 1855676, 2127122, 2429577
Offset: 1
a(1)=1 because there are 2 ways to form a regular tetrahedron using vertices of the unit cube: Either [(0,0,0),(0,1,1),(1,0,1),(1,1,0)] or [(1,1,1),(1,0,0),(0,1,0),(0,0,1)].
- E. J. Ionascu, Regular tetrahedra whose vertices have integer coordinates. Acta Math. Univ. Comenian. (N.S.) 80 (2011), no. 2, 161-170; (Acta Mathematica Universitatis Comenianae) MR2835272 (2012h:11044).
- Eugen J. Ionascu, Table of n, a(n) for n = 1..100
- Eugen J. Ionascu, A characterization of regular tetrahedra in Z^3, Journal of Number Theory, Volume 129, Issue 5, May 2009, pp. 1066-1074.
- Eugen J. Ionascu, Counting all regular tetrahedra in {0,1,...,n}^3, arXiv:0912.1062 [math.NT], 2009.
- Eugen J. Ionascu, Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, pp. 138-145.
- Eugen J. Ionascu, Regular tetrahedra whose vertices have integer coordinates, Acta Mathematica Universitatis Comenianae, Vol. LXXX, 2 (2011) pp. 161-170.
- Eugen J. Ionascu and R. A. Obando, Cubes in {0,1,...,N}^3, INTEGERS, 12A (2012), #A9. - From _N. J. A. Sloane_, Feb 05 2013
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).
A103501
(1/8)*number of equilateral 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
1, 10, 46, 158, 431, 974, 2022, 3837, 6777, 11263, 17947, 27541, 40835, 58904, 83081, 114543, 155232, 206901, 271573, 351583, 449833, 569225, 712847, 884408, 1088136, 1328616, 1610007, 1937077, 2315434, 2750476, 3250073, 3820925, 4469597
Offset: 1
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
A190019
Number of acute triangles on an n X n grid (or geoboard).
Original entry on oeis.org
0, 0, 8, 80, 404, 1392, 3880, 9208, 19536, 38096, 69288, 119224, 196036, 310008, 474336, 705328, 1023216, 1451904, 2020232, 2762848, 3719420, 4937200, 6469424, 8378184, 10734664, 13618168, 17119288, 21338760, 26390452, 32400592, 39508656, 47870200, 57655752
Offset: 1
Cf.
A103429 (analogous problem on a 3-dimensional grid).
Showing 1-8 of 8 results.
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