A343066 -id:A343066 - cp's OEIS frontend

A343063 Primitive triples (a, b, c) for integer-sided triangles whose angle B = 2*C.

5, 6, 4, 7, 12, 9, 9, 20, 16, 11, 30, 25, 13, 42, 36, 15, 56, 49, 16, 15, 9, 17, 72, 64, 19, 90, 81, 21, 110, 100, 23, 132, 121, 24, 35, 25, 25, 156, 144, 27, 182, 169, 29, 210, 196, 31, 240, 225, 32, 63, 49, 33, 28, 16, 33, 272, 256, 35, 306, 289, 37, 342, 324, 39, 40, 25, 39, 380, 361, 40, 99, 81, 41, 420, 400, 43, 462, 441

Offset: 1

Bernard Schott, Apr 04 2021

This sequence is inspired by the problem of French Baccalauréat Mathématiques at Lyon in 1937 (see link).
The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
This metric relation is equivalent to a = m^2 - k^2, b = m * k, c = k^2, gcd(m,k) = 1 and k < m < 2k; hence every c is a square number.
When A <> 45° and A <> 72°, table below shows there exist these 3 possible inequalities: c < b < a, c < a < b, a < c < b.
   ------------------------------------------------------------------------
   | A | 180 |   decr.  |   72    |   decr.   |   45    |   decr.   |  0  |
   ------------------------------------------------------------------------
   | B |  0  |   incr.  |   72    |   incr.   |   90    |   incr.   | 120 |
   ------------------------------------------------------------------------
   | C |  0  |   incr.  |   36    |   incr.   |   45    |   incr.   |  60 |
   ------------------------------------------------------------------------
   | < | No | c < b < a | c < b=a | c < a < b | c=a < b | a < c < b |  No |
   ------------------------------------------------------------------------
where 'No' means there is no such corresponding triangle.
If (A,B,C) = (72,72,36) then a = b = c * (1+sqrt(5))/2 and isosceles ABC is not an integer-sided triangle.
If (A,B,C) = (45,90,45) then ABC is isosceles rectangle in B, so a = c with b = a*sqrt(2) and ABC is not an integer-sided triangle.

			The smallest such triangle is (5, 6, 4), of type c < a < b with 4*(5+4) = 6^2.
The 2nd triple is (7, 12, 9) of type a < c < b with 9*(7+9) = 16^2.
The 7th triple (16, 15, 9) is the first of type c < b < a with 9*(16+9) = 15^2.
The table begins:
   5,  6,  4;
   7, 12,  9;
   9, 20, 16;
  11, 30, 25,
  13, 42, 36;
  15, 56, 49;
  16, 15,  9;
  17, 72, 64;
  ...

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-336 p. 178, André Desvigne.

APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Cf. A335893 (similar for A < B < C in arithmetic progression).

Cf. A343064 (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).

for a from 2 to 60 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if igcd(a,sqrt(d),c)=1 and issqr(d) and abs(a-c)

A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 33, 35, 37, 39, 39, 40, 41, 43, 45, 47, 48, 49, 51, 51, 53, 55, 56, 56, 57, 57, 59, 61, 63, 64, 65, 67, 69, 69, 71, 72, 72, 73, 75, 75, 77, 79, 80, 81, 83, 85, 85, 87, 87, 88, 88, 89, 91, 93, 93, 95, 95, 96, 97, 99

Offset: 1

Bernard Schott, Apr 10 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
In this case, the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
Equivalently, length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

			According to inequalities between a, b, c, there exist 3 types of such triangles:
   c < a < b for the smallest side a = 5 and triple (5, 6, 4).
The first side a for which there exist two distinct triangles with B = 2C is for a = 33 with the two other types of examples,
   c < b < a with triple (33, 28, 16),
   a < c < b with triple (33, 272, 256).

APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Cf. A353619 (similar, but with B = 3*C).
Cf. A343063 (triples), this sequence (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).
Cf. A106505 (sides a without repetition), A106506 (sides a sorted on perimeter).

for a from 2 to 100 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if issqr(d) and  igcd(a,sqrt(d),c)=1 and abs(a-c)

a(n) = A343063(n, 1).

A343065 Side b of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

6, 12, 20, 30, 42, 56, 15, 72, 90, 110, 132, 35, 156, 182, 210, 240, 63, 28, 272, 306, 342, 40, 380, 99, 420, 462, 506, 552, 143, 600, 70, 650, 702, 756, 45, 195, 88, 812, 870, 930, 992, 255, 1056, 1122, 130, 1190, 1260, 77, 323, 1332, 154, 1406, 1482, 1560, 399, 1640, 1722, 66, 1806, 208, 1892, 117, 483, 1980, 2070, 238

Offset: 1

Bernard Schott, Apr 11 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
This sequence is not increasing because a(7) = 15 < a(6) = 56 (A106430).
If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
Equivalently, length of side opposite to the greater of the two angles, one being the double of the other.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

			According to inequalities between a, b, c, there exist 3 types of such triangles:
c < a < b for the first triple (5, 6, 4) with b = 6.
c < b < a for the second triple (16, 15, 9) with b = 15.
a < c < b for the seventh triple (7, 12, 9) with b = 12.

APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Cf. A335895 (similar for A < B < C in arithmetic progression).
Cf. A343063 (triples), A343064 (side a), A343066 (side c), A343067 (perimeter).
Cf. A106420 (sides b sorted on perimeter), A106430 (sides b in increasing order).

for a from 2 to 100 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if issqr(d) and igcd(a,sqrt(d),c)=1 and abs(a-c)

a(n) = A343063(n, 2).

A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

15, 28, 45, 66, 91, 120, 40, 153, 190, 231, 276, 84, 325, 378, 435, 496, 144, 77, 561, 630, 703, 104, 780, 220, 861, 946, 1035, 1128, 312, 1225, 170, 1326, 1431, 1540, 126, 420, 209, 1653, 1770, 1891, 2016, 544, 2145, 2278, 299, 2415, 2556, 198, 684, 2701, 350, 2850, 3003, 3160

Offset: 1

Bernard Schott, Apr 15 2021

nonn

The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide then in increasing order of side b.
This sequence is nonincreasing: a(7) = 40 < a(6) = 120.
If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
As the metric relation is equivalent to a = m^2 - k^2, b = m*k, c = k^2, with gcd(m,k) = 1 and k < m < 2k, so all terms are of the form m^2 + m*k = m * (m+k) with gcd(m,k) = 1 and k < m < 2k. These perimeters are in increasing order in A106499.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a(1) = 15 with c < a < b for the first triple (5, 6, 4);
a(7) = 40 with c < b < a for the seventh triple (16, 15, 9);
a(8) = 153 with a < c < b for the eighth triple (17, 72, 64).

Cf. A335897 (similar for A < B < C in arithmetic progression).

Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343066 (side c), A106499 (perimeters in increasing order).

for a from 2 to 100 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if issqr(d) and igcd(a, sqrt(d), c)=1 and abs(a-c)

lista(nn) = {for (a = 2, nn, for (c = 3, a^2\2, my(d = c*(a+c)); if (issquare(d) && (gcd([a, sqrtint(d), c])==1) && (abs(a-c)Michel Marcus, May 12 2022

a(n) = A343063(n, 1) + A343063(n, 2) + A343063(n, 3).

a(n) = A343064(n) + A343065(n) + A343066(n).

A106505 Ordered and uniqued length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105, 107, 109, 111, 112

Offset: 1

Lekraj Beedassy, May 04 2005

nonn

The terms are proposed without repetition. For example, there exist two such triangles with a length of side = 33. They correspond respectively to s^2 - r^2 = 33 (see formula) with (r, s) = (4, 7) and sides (33, 28, 16), and the other triangle with (r, s) = (16, 17) and sides (33, 272, 256). Lengths = 39, 51, 57, 69, 75, ... correspond to two distinct triangles ... The lengths of these sides are proposed with repetition in A343064. - Bernard Schott, Apr 22 2021

Cf. A106499-A106506, A106410, A106420, A106430, A321499.

Cf. A343063, A343064, A343065, A343066, A343067.

Values s^2 - r^2, where r

Conjecture: for n>2, a(n+5) = a(n) + 8. - Ralf Stephan, Nov 16 2010.

Empirical g.f.: -x*(x^7+x^6+3*x^5-2*x^4-2*x^3-2*x^2-2*x-5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Oct 05 2013

Extended by Ray Chandler, May 09 2005

A353621 Side c of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

Original entry on oeis.org

8, 27, 64, 125, 125, 343, 343, 216, 343, 729, 343, 512, 729, 512, 1331, 729, 1728, 1331, 729, 1000, 1331, 2197, 1728, 1000, 1331, 2197, 1331, 1331, 2197, 3375, 2197, 4096, 3375, 1728, 2744, 2197, 2197, 4913, 4096, 2197, 2744, 4913, 3375, 6859, 2744, 4913, 4096, 6859, 3375, 4913, 8000

Offset: 1

Bernard Schott, May 07 2022

nonn

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of side c; hence, this sequence of sides c is not increasing.
In the case B = 3*C, the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
Equivalently, length of side opposite to the angle that is one-third of another one, for primitive integer-sided triangles.
All terms are cubes >= 8 (A000578). More generally, when c is the side of a primitive integer-sided triangles (a, b, c) whose angle B = m*C, then c = k^m, for some k >= 2.
Note that side c is never the largest side of the triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A353618.

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the middle side c = 8 of the first triple (3, 10, 8).
c < a < b with the smallest side c = 27 of the 2nd triple (35, 48, 27).
c < b < a with the smallest side c = 216 of the 8th triple (539, 510, 216), the first of this type.
The smallest side c for which there exist two distinct triangles with B = 3*C is for a(4) = a(5) = 125, and these sides c belong respectively to triples (112, 195, 125) and (279, 280, 125).

The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Cf. A353618 (triples), A353619 (side a), A353620 (side b), this sequence (side c), A353622 (perimeter).
Cf. A343066 (similar, but with B = 2*C).
Cf. A000578.

for b from 4 to 9000  do
  for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a,b,q)=1 and (b-q^3) < a and a < b+q^3 then print(q^3); end if;
end do;
end do;

a(n) = A353618(n, 3).

Showing 1-6 of 6 results.

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A343063 Primitive triples (a, b, c) for integer-sided triangles whose angle B = 2*C.

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A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.

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A343065 Side b of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

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A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

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A106505 Ordered and uniqued length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.

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A353621 Side c of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

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