A106430 Ordered and uniqued length of side opposite the greater of the two angles, one being the double of the other, of a primitive integer-sided triangle.
6, 12, 15, 20, 28, 30, 35, 40, 42, 45, 56, 63, 66, 70, 72, 77, 84, 88, 90, 91, 99, 104, 110, 117, 120, 126, 130, 132, 143, 144, 153, 154, 156, 165, 170, 176, 182, 187, 190, 195, 198, 204, 208, 209, 210, 220, 221, 228, 231, 234, 238, 240, 247, 255, 260, 266, 272, 273, 276, 285, 286, 299
Offset: 1
Keywords
Links
- Albert Lau, Table of n, a(n) for n = 1..10539
Programs
-
Mathematica
aMax = 300; Select[Sequence @@@ Table[{m (m + n), n (m + n), m n}, {m, Sqrt[aMax]}, {n, Min[m - 1, aMax/m - m]}], GCD @@ # == 1 &] // Sort; %[[;; , 1]] (* Albert Lau, Sep 19 2014 *)
Formula
Values r*s, where r
The other 2 sides are s^2 and r^2-s^2. - Albert Lau, Sep 19 2014
A106499 Ordered perimeter of primitive integer-sided triangle one of whose angles is twice another.
15, 28, 40, 45, 66, 77, 84, 91, 104, 120, 126, 144, 153, 170, 187, 190, 198, 209, 220, 228, 231, 260, 273, 276, 286, 299, 312, 322, 325, 345, 350, 378, 390, 400, 420, 432, 435, 442, 459, 464, 476, 493, 496, 510, 522, 527, 544, 551, 558, 561, 570, 589, 608
Offset: 1
Keywords
Comments
Links
- Jean-François Alcover, Table of n, a(n) for n = 1..869
Programs
-
Mathematica
s[p_] := Solve[1 < r < s < 2r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers]; Reap[For[p = 1, p <= 10000, p++, sp = s[p]; If[sp =!= {}, Sow[p]; Print[p, " ", {r s, s^2 - r^2, r^2} /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 06 2020 *)
A106420 Length of side opposite the greater of the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter (A106499).
6, 12, 15, 20, 30, 28, 35, 42, 40, 56, 45, 63, 72, 70, 66, 90, 77, 88, 99, 84, 110, 91, 104, 132, 117, 130, 143, 126, 156, 120, 154, 182, 165, 144, 195, 176, 210, 153, 170, 208, 187, 204, 240, 221, 198, 238, 255, 190, 234, 272, 209, 228, 247, 220, 266, 306, 285
Offset: 1
Keywords
Programs
-
Mathematica
sol[p_] := Solve[1 < r < s < 2 r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers]; Reap[For[p = 1, p <= 1000, p++, sp = sol[p]; If[sp =!= {}, Print[r s /. sp[[1]]]; Sow[r s /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)
A106410 Length of side opposite the lesser of the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter(A106499).
4, 9, 9, 16, 25, 16, 25, 36, 25, 49, 25, 49, 64, 49, 36, 81, 49, 64, 81, 49, 100, 49, 64, 121, 81, 100, 121, 81, 144, 64, 121, 169, 121, 81, 169, 121, 196, 81, 100, 169, 121, 144, 225, 169, 121, 196, 225, 100, 169, 256, 121, 144, 169, 121, 196, 289, 225, 169, 256, 121
Offset: 1
Keywords
Programs
-
Mathematica
sol[p_] := Solve[1 < r < s < 2 r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers]; Reap[For[p = 1, p <= 1000, p++, sp = sol[p]; If[sp =!= {}, Print[r^2 /. sp[[1]]]; Sow[r^2 /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)
Formula
a(n) = A106500(n)^2.
A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.
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
Keywords
Comments
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.
Examples
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).
Links
Crossrefs
Programs
-
Maple
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)
Formula
a(n) = A343063(n, 1).
A106500 Values r associated with A106499.
2, 3, 3, 4, 5, 4, 5, 6, 5, 7, 5, 7, 8, 7, 6, 9, 7, 8, 9, 7, 10, 7, 8, 11, 9, 10, 11, 9, 12, 8, 11, 13, 11, 9, 13, 11, 14, 9, 10, 13, 11, 12, 15, 13, 11, 14, 15, 10, 13, 16, 11, 12, 13, 11, 14, 17, 15, 13, 16, 11, 17, 18, 13, 17, 13, 16, 19, 17, 12, 15, 13, 19, 14, 17, 20, 15, 13, 16, 19
Offset: 1
Keywords
Programs
-
Mathematica
s[p_] := Solve[1 < r < s < 2r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers]; Reap[For[p = 1, p <= 2000, p++, sp = s[p]; If[sp =!= {}, Print[r /. sp[[1]] ]; Sow[r /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 06 2020 *)
Extensions
Extended by Ray Chandler, May 09 2005
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.
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
Keywords
Comments
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
Crossrefs
Formula
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
Extensions
Extended by Ray Chandler, May 09 2005
A106501 Values s associated with A106499.
3, 4, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 11, 10, 11, 11, 11, 12, 11, 13, 13, 12, 13, 13, 13, 14, 13, 15, 14, 14, 15, 16, 15, 16, 15, 17, 17, 16, 17, 17, 16, 17, 18, 17, 17, 19, 18, 17, 19, 19, 19, 20, 19, 18, 19, 20, 19, 21, 19, 19, 21, 20, 22, 21, 20, 21, 23, 22, 23, 21, 23, 22
Offset: 1
Keywords
Extensions
Extended by Ray Chandler, May 09 2005
A106502 Shortest side of primitive integer-sided triangle having an angle twice another, sorted on perimeter (A106499).
4, 7, 9, 9, 11, 16, 24, 13, 25, 15, 25, 32, 17, 49, 36, 19, 49, 57, 40, 49, 21, 49, 64, 23, 81, 69, 48, 81, 25, 64, 75, 27, 104, 81, 56, 121, 29, 81, 100, 87, 121, 144, 31, 120, 121, 93, 64, 100, 155, 33, 121, 144, 169, 121, 165, 35, 136, 169, 105, 121, 72, 37, 169, 111, 169
Offset: 1
Keywords
Comments
Triple is (A106502,106503,106504).
Extensions
Corrected and extended by Ray Chandler, May 09 2005
A106503 Middle side of primitive integer-sided triangle having an angle twice another, sorted on perimeter (A106499).
5, 9, 15, 16, 25, 28, 25, 36, 39, 49, 45, 49, 64, 51, 66, 81, 72, 64, 81, 84, 100, 91, 104, 121, 88, 100, 121, 115, 144, 120, 121, 169, 121, 144, 169, 135, 196, 153, 170, 169, 168, 145, 225, 169, 198, 196, 225, 190, 169, 256, 209, 217, 192, 220, 196, 289, 225, 231
Offset: 1
Keywords
Extensions
Corrected and extended by Ray Chandler, May 09 2005
Comments