A201223 Primitive Eisenstein triples (a,b,c) listed as groups of three in order of increasing b.
3, 8, 7, 5, 8, 7, 7, 15, 13, 8, 15, 13, 5, 21, 19, 16, 21, 19, 11, 35, 31, 24, 35, 31, 7, 40, 37, 33, 40, 37, 13, 48, 43, 35, 48, 43, 16, 55, 49, 39, 55, 49, 9, 65, 61, 56, 65, 61, 32, 77, 67, 45, 77, 67, 17, 80, 73, 63, 80, 73, 40, 91, 79, 51, 91, 79, 11, 96, 91
Offset: 1
Keywords
A264826 Primitive Eisenstein triples: (a,b,c) in lexicographic order such that a^2 + b^2 - a*b - c^2 = 0, a < b < c, and gcd(a, b) = 1.
3, 7, 8, 5, 7, 8, 5, 19, 21, 7, 13, 15, 7, 37, 40, 8, 13, 15, 9, 61, 65, 11, 31, 35, 11, 91, 96, 13, 43, 48, 13, 127, 133, 15, 169, 176, 16, 19, 21, 16, 49, 55, 17, 73, 80, 17, 217, 225, 19, 91, 99, 19, 271, 280, 21, 331, 341, 23, 133, 143, 23, 397, 408
Offset: 1
Comments
The sides of a primitive 60-degree integer triangle.
Links
- Colin Barker, Table of n, a(n) for n = 1..9999
- Eric Weisstein's World of Mathematics, Pythagorean Triple
- Wikipedia, Eisenstein triple
Programs
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PARI
pt60(a) = { my(L=List(), n=-3*a^2, f, g, b, c); fordiv(n, f, g=n\f; if(f>g && (g+f)%2==0 && (f-g)%4==0, b=(f-g)\4; c=((f+g)\2+a)\2; if(c>0 && a
A350013 Number of integer-sided triangles with one side having length n and an adjacent angle of 60 degrees.
1, 1, 2, 1, 3, 2, 3, 4, 3, 3, 3, 2, 3, 3, 7, 6, 3, 3, 3, 3, 7, 3, 3, 7, 5, 3, 4, 3, 3, 7, 3, 8, 6, 3, 10, 3, 3, 3, 6, 11, 3, 7, 3, 3, 10, 3, 3, 12, 5, 5, 6, 3, 3, 4, 10, 10, 6, 3, 3, 7, 3, 3, 10, 10, 10, 6, 3, 3, 6, 10, 3, 10, 3, 3, 11, 3, 10, 6, 3, 18, 5, 3, 3, 7, 9, 3, 6, 10, 3, 10, 10
Offset: 1
Comments
All terms are greater than or equal to 1 because a triangle with side lengths {n, n, n} is equilateral and has an adjacent angle of 60 degrees.
Number of possible integer solutions to the equation n^2 + x^2 - nx = y^2.
x <= n^2 and y <= n^2. - Seiichi Manyama, Dec 09 2021
From David A. Corneth, Dec 10 2021: (Start)
Solving n^2 + x^2 - nx = y^2 for x using the quadratic formula gives x = (n +- sqrt(4*y^2 - 3*n^2)) / 2.
So we need sqrt(4*y^2 - 3*n^2) to be an integer, say k, i.e., sqrt(4*y^2 - 3*n^2) = k.
Squaring gives 4*y^2 - 3*n^2 = k^2, i.e., (2y - k)*(2y + k) = 4*y^2 - k^2 = 3*n^2
Checking divisors d of 3*n^2 gives all candidates for y = (d + 3*n^2/d)/4 and x = (n +- sqrt(4*y^2 - 3*n^2)) / 2 which must be positive. (End)
Examples
For n = 8, there are 4 possible integer triangles with side length 8 and adjacent angle 60 degrees. Their side lengths are {8, 3, 7}, {8, 5, 7}, {8, 8, 8}, {8, 15, 13}.
Links
- David A. Corneth, Table of n, a(n) for n = 1..10000
Programs
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PARI
a(n) = sum(x=1, n^2, issquare(x^2 - n * x + n^2)); \\ David A. Corneth, Dec 09 2021
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PARI
a(n) = { my(n23 = 3*n^2, d = divisors(n23), res = 0); for(i = 1, (#d + 1)\2, y = (d[i] + n23/d[i])/4; if(denominator(y) == 1, x = (n + sqrtint(4*y^2 - n23))/2; if(denominator(x) == 1, res++ ); x = (n - sqrtint(4*y^2 - n23))/2; if(x > 0 && denominator(x) == 1, res++ ); ) ); res } \\ faster than above \\ David A. Corneth, Dec 10 2021
Extensions
More terms from David A. Corneth, Dec 09 2021
A242039 List of integers b such that (a1,b,c1) and (a2,b,c2) are primitive Eisenstein triples, max(a1,b,c1,a2,c2)=b, and a1,c1,a3,c3 are distinct.
280, 1144, 1155, 1680, 1768, 1976, 2145, 2584, 2805, 3003, 3128, 3315, 3360, 3400, 3496, 3705, 3800, 4095, 4600, 4845, 5005, 5280, 5336, 5355, 5704, 5720, 5800, 5865, 5985, 6160, 6200, 6240, 6545, 6555, 6783, 6864, 7192, 7280, 7315, 7400, 7735, 8120, 8265, 8584, 8645, 8680, 8835, 8855, 9176, 9177, 9240, 9360, 9512, 9976
Offset: 1
Keywords
Comments
For Eisenstein triple see A121992.
Examples
280 is in the list because (93,280,247) and (19,280,271) are primitive Eisenstein triples and 280 is the largest side and no other side is equal. Consider (3,8,7) and (5,8,7), 8 is not in the list because 7 appear in both triple.
Programs
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Mathematica
max = 2000; data = Do[Sqrt[-3 a^2 + 4 c^2] // If[IntegerQ[#] && GCD[a, c] == 1, {a, (a + #)/2, c} // Sow] &, {a, max}, {c, Sqrt[3]/2 a // Ceiling, a - 1}] // Reap // Last // Last; Select[data[[;; , 1]] // Tally, #[[2]] > 1 &][[;; , 1]]
Comments
Examples
Links
Crossrefs
Programs
Mathematica
x = {}; For[b = 1, b <= 77, b++, For[c = 1, c < b, c++, For[a = 1, a < c, a++, {If[(a^2 - a*b + b^2 == c^2) && (GCD[a, b, c] == 1), AppendTo[x, {a, b, c}]]}]]]; Flatten[x]