cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A336328 Primitive triples for integer-sided triangles with A < B < C < 2*Pi/3 and such that FA + FB + FC is an integer where F is the Fermat point of the triangle.

Original entry on oeis.org

57, 65, 73, 73, 88, 95, 43, 147, 152, 127, 168, 205, 97, 185, 208, 111, 221, 280, 49, 285, 296, 95, 312, 343, 296, 315, 361, 152, 343, 387, 323, 392, 407, 147, 377, 437, 285, 464, 469, 255, 343, 473, 247, 408, 485, 469, 589, 624, 403, 725, 728, 871, 901, 931
Offset: 1

Views

Author

Bernard Schott, Jul 17 2020

Keywords

Comments

Inspired by Project Euler, Problem 143 (see link).
The triples are displayed in increasing order of largest side c, and if largest sides coincide then by increasing order of the middle side b; so, each triple (a, b, c) is in increasing order.
If one angle of the triangle, for example C, is >= 2*Pi/3 then the Fermat point F is this vertex C, so, FA + FB + FC becomes CA + CB, while when all angles are < 2*Pi/3, then the Fermat point is inside the triangle (see link Fermat points), this last condition means that c^2 < a^2 + a*b + b^2.
As a < b < c, then FA > FB > FC.
If FA + FB + FC = d, then we have this "beautifully symmetric equation" between a, b, c and d: 3*(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2 (see Martin Gardner).
Equivalently: if a point M is inside an equilateral triangle A'B'C' and integer distances to vertices are MA' = a = A072054(n), MB' = b = A072053(n), MC' = c = A072052(n), then the side of this equilateral triangle A'B'C' is equal to d = FA + FB + FC = A061281(n) where F is the Fermat point of the triangle ABC with sides (a,b,c) (see Martin Gardner).
+-----+-----+-----+-----------+-----------+-----------+-----+-------+
| a | b | c | FA | FB | FC | d | a+b+c |
+-----------+-----+-----------+-----------+-----------+-----+-------+
| 57 | 65 | 73 | 325/7 | 264/7 | 195/7 | 112 | 195 |
| 73 | 88 | 95 | 440/7 | 325/7 | 264/7 | 147 | 256 |
| 43 | 147 | 152 | 5016/37 | 1064/37 | 765/37 | 185 | 342 |
| 127 | 168 | 205 | 39360/283 | 27265/283 | 13464/283 | 283 | 500 |
| 97 | 185 | 208 | 14800/91 | 6528/91 | 3515/91 | 273 | 490 |
| 111 | 221 | 280 | 70720/331 | 34200/331 | 4641/331 | 331 | 612 |
| 49 | 285 | 296 | 91200/331 | 12376/331 | 5985/331 | 331 | 630 |
| 95 | 312 | 343 | 3864/13 | 1015/13 | 360/13 | 403 | 750 |
| 296 | 315 | 361 | 9405/43 | 8512/43 | 6120/43 | 559 | 972 |
| 152 | 343 | 387 | 30429/97 | 11520/97 | 5096/97 | 485 | 882 |
.....................................................................
From the previous table, we observe that every FA, FB, FC is a fraction while FA + FB + FC = d is an integer (A336329). Jinyuan Wang has found that the 37th triple is the first for which the common denominator of these fractions is 1 (A351477).

Examples

			The table begins:
   57,  65,  73;
   73,  88,  95;
   43, 147, 152;
  127, 168, 205;
   97, 185, 208;
  111, 221, 280;
   49, 285, 296;
  .............
For first triple (57, 65, 73) and corresponding d = FA + FB + FC = 325/7 + 264/7 + 195/7 = 112, relation gives: 3*(57^4 + 65^4 + 73^4 + 112^4) = (57^2 + 65^2 + 73^2 + 112^2)^2 = 642470409.

References

  • Martin Gardner, Mathematical Circus, Elegant triangles, First Vintage Books Edition, 1979, p. 65.

Links

Crossrefs

Cf. A336329 (FA + FB + FC), A336330 (smallest side), A336331 (middle side), A336332 (largest side), A336333 (perimeter), A351801 (FA numerator), A351802 (FB numerator), A351803 (FC numerator), A351477 (common denominator of FA, FB, FC), A351476 (other FA + FB + FC).
Cf. A333391 (with isogonic center).
Cf. A061281, A072052, A072053, A072054.

Formula

If FA + FB + FC = d, then
d = sqrt(((a^2 + b^2 + c^2)/2) + (1/2) * sqrt(6*(a^2*b^2 + b^2*c^2 + c^2*a^2) - 3*(a^4 + b^4 + c^4))), or,
d^2 = (1/2) * (a^2 + b^2 + c^2) + 2 * S * sqrt(3) where S = area of triangle ABC.

A336332 Largest side, in increasing order, of primitive integer-sided triangles with A < B < C < 2*Pi/3 and such that FA + FB + FC is an integer where F is the Fermat point of the triangle.

Original entry on oeis.org

73, 95, 152, 205, 208, 280, 296, 343, 361, 387, 407, 437, 469, 473, 485, 624, 728, 931, 1016, 1273, 1311, 1313, 1368, 1387, 1443, 1457, 1463, 1469, 1477, 1519, 1560, 1591, 1687, 1895, 2015, 2045, 2045, 2085, 2197, 2231, 2289, 2347, 2363, 2416, 2465, 2553, 2728, 2821, 2923
Offset: 1

Views

Author

Bernard Schott, Jul 20 2020

Keywords

Comments

Inspired by Project Euler, Problem 143 (see link).
This sequence is increasing because triples are in increasing order of largest side.
For the corresponding primitive triples and miscellaneous properties and references, see A336328.
If FA + FB + FC = d, then we have this "beautifully symmetric equation" between a, b, c and d (see Martin Gardner): 3*(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.

Examples

			a(36) = a(37) = 2045 is the smallest largest side that appears twice because:
   (1023, 1387, 2045) is a triple with FA+FB+FC = 2408, and
   (1051, 1744, 2045) is a triple with FA+FB+FC = 2709.

References

  • Martin Gardner, Mathematical Circus, Elegant triangles, First Vintage Books Edition, 1979, p. 65

Links

Crossrefs

Cf. A336328 (triples), A336329 (FA + FB + FC), A336330 (smallest side), A336331 (middle side), this sequence (largest side), A336333 (perimeter).
Cf. A072052 (largest sides: primitives and multiples), A333391.

Formula

a(n) = A336328(n, 3).

A061281 Side of n-th equilateral triangle enclosing at least one point located at integer distances from the vertices.

Original entry on oeis.org

112, 147, 185, 224, 273, 283, 294, 331, 331, 336, 370, 403, 441, 448, 485, 520, 546, 555, 559, 560, 566, 588, 592, 637, 645, 662, 662, 672, 691, 735, 740, 784, 806, 819, 849, 882, 896, 925, 965, 970, 993, 993, 1008, 1029, 1040, 1047, 1092, 1110, 1118, 1120, 1132
Offset: 1

Views

Author

Lekraj Beedassy, May 21 2001

Keywords

Comments

The equation has many other integer solutions, such as {3,5,7,8}; most of these describe points that lie on the edge of the triangle. - David Wasserman, Jun 10 2002. See A089025.

Examples

			The solution (97,185,208,273) of the triangle equation gives rise to the value 273 as the 5th equilateral triangle associated with an interior point at integer distances from the vertices.

References

  • M. Gardner, Mathematical Circus, Alfred A. Knopf, 1979, p. 65.
  • L. Pianaro, Pierre Est Encore Perdu, Jouer Jeux Mathematiques, No. 18, Oct 1995, published by French Federation of Mathematics Games.

Crossrefs

Cf. A072052, A072053, A072054, A089025.

Formula

a(n) is the largest term in the n-th quadruple (a, b, c, d) satisfying the triangle equation 3*(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.

Extensions

More terms from David Wasserman, Jun 10 2002
More terms from Jinyuan Wang, Jul 20 2020

A072053 Median of 3 integer distances referred to in A061281.

Original entry on oeis.org

65, 88, 147, 130, 185, 168, 176, 221, 285, 195, 294, 312, 264, 260, 343, 377, 370, 441, 315, 325, 336, 352, 343, 408, 392, 442, 570, 390, 464, 440, 588, 455, 624, 555, 504, 528, 520, 735, 589, 686, 663, 855, 585, 616, 754, 725, 740, 882, 630, 650, 672, 704, 686
Offset: 1

Views

Author

David Wasserman, Jun 10 2002

Keywords

Examples

			a(2) = 88 because the equilateral triangle with side length A061281(2) = 147 has an interior point whose distances from the vertices are 95, 88 and 73.

Crossrefs

Cf. A061281, A072052, A072054.

Extensions

More terms from Jinyuan Wang, Jul 20 2020

A072054 Smallest of 3 integer distances referred to in A061281.

Original entry on oeis.org

57, 73, 43, 114, 97, 127, 146, 111, 49, 171, 86, 95, 219, 228, 152, 147, 194, 129, 296, 285, 254, 292, 255, 247, 323, 222, 98, 342, 285, 365, 172, 399, 190, 291, 381, 438, 456, 215, 469, 304, 333, 147, 513, 511, 294, 403, 388, 258, 592, 570, 508, 584, 510, 285
Offset: 1

Views

Author

David Wasserman, Jun 10 2002

Keywords

Examples

			a(3) = 43 because the equilateral triangle with side length A061281(3) = 185 has an interior point whose distances from the vertices are 152, 147 and 43.

Crossrefs

Cf. A061281, A072052, A072053.

Extensions

More terms from Jinyuan Wang, Jul 20 2020
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