A229849 -id:A229849 - cp's OEIS frontend

A229858 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of A.

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Colin Barker, Oct 06 2013

A229859 gives the values of B, and A050931 gives the values of C.

This sequence contains every integer larger than 8. - Nathaniel Johnston, Oct 06 2013

			12 appears in the sequence because there exists a 120-degree triangle with sides 12, 20 and 28.

Wikipedia, Integer triangle
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A050931, A229849, A229859.

\\ Gives values of A not exceeding amax.
\\ e.g. t120a(20) gives [3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
t120a(amax) = {
  v=pt120a(amax);
  s=[];
  for(i=1, #v,
    for(m=1, amax\v[i],
      if(v[i]*m<=amax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of A not exceeding amax in primitive triangles.
\\ e.g. pt120a(20) gives [3, 5, 7, 9, 11, 13, 15, 16, 17, 19]
pt120a(amax) = {
  s=[];
  for(m=1, (amax-1)\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        a=m*m-n*n;
        b=n*(2*m+n);
        if(a>b, a=b);
        if(a<=amax, s=concat(s, a))
      )
    )
  );
  vecsort(s,,8)
}

a(n) = n+4 for n>4.
a(n) = 2*a(n-1)-a(n-2) for n>6.
G.f.: -x*(x^5-x^4+x^2+x-3) / (x-1)^2.

A264827 (a,b,c) in lexicographic order such that a^2 + b^2 + a*b - c^2 = 0 with a < b < c and gcd(a, b) = 1.

Original entry on oeis.org

3, 5, 7, 5, 16, 19, 7, 8, 13, 7, 33, 37, 9, 56, 61, 11, 24, 31, 11, 85, 91, 13, 35, 43, 13, 120, 127, 15, 161, 169, 16, 39, 49, 17, 63, 73, 17, 208, 217, 19, 80, 91, 19, 261, 271, 21, 320, 331, 23, 120, 133, 23, 385, 397, 24, 95, 109, 25, 143, 157

Offset: 1

Colin Barker, Nov 26 2015

The sides of a primitive 120-degree integer triangle.

			Triples (a,b,c) begin:
  3,  5,  7;
  5, 16, 19;
  7,  8, 13;
  7, 33, 37;
  9, 56, 61;
  ...

Colin Barker, Table of n, a(n) for n = 1..9999
Wikipedia, Eisenstein triple

Cf. A002476, A229849, A229858, A229859, A264826.

pt120(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,
      c=(f-g)\4; b=((f+g)\2-a)\2;
      if(b>0 && a

A229859 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of B.

Original entry on oeis.org

5, 8, 10, 15, 16, 20, 24, 25, 30, 32, 33, 35, 39, 40, 45, 48, 50, 51, 55, 56, 57, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 85, 88, 90, 91, 95, 96, 99, 100, 102, 104, 105, 110, 112, 114, 115, 117, 120, 125, 126, 128, 130, 132, 135, 136, 140, 143, 144, 145

Offset: 1

Colin Barker, Oct 06 2013

nonn

A229858 gives the values of A, and A050931 gives the values of C.

			20 appears in the sequence because there exists a 120-degree triangle with sides 12, 20 and 28.

Wikipedia, Integer triangle

Cf. A050931, A229849, A229858.

\\ Gives values of B not exceeding bmax.
\\ e.g. t120b(40) gives [5, 8, 10, 15, 16, 20, 24, 25, 30, 32, 33, 35, 39, 40]
t120b(bmax) = {
  v=pt120b(bmax);
  s=[];
  for(i=1, #v,
    for(m=1, bmax\v[i],
      if(v[i]*m<=bmax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of B not exceeding bmax in primitive triangles.
\\ e.g. pt120b(40) gives [5, 8, 16, 24, 33, 35, 39]
pt120b(bmax) = {
  s=[];
  for(m=1, (bmax-1)\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        a=m*m-n*n;
        b=n*(2*m+n);
        if(a>b, b=a);
        if(b<=bmax, s=concat(s, b))
      )
    )
  );
  vecsort(s,,8)
}

A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

Original entry on oeis.org

3, 7, 5, 11, 7, 13, 16, 9, 32, 17, 40, 11, 19, 55, 40, 24, 13, 23, 65, 69, 56, 25, 75, 15, 104, 32, 56, 29, 17, 87, 85, 119, 31, 72, 93, 64, 144, 19, 95, 133, 40, 136, 35, 105, 21, 105, 37, 111, 185, 88, 152, 176, 23, 80, 115, 161, 41, 123, 240, 48, 205, 240, 43, 25, 129, 175, 215, 88

Offset: 1

Bernard Schott, Sep 23 2022

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the a's.
For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions a of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, a is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with a = u^2 - v^2.
This sequence is not increasing. For example, a(2) = 7 for triangle with largest side = 13 while a(3) = 5 for triangle with largest side = 19.
Differs from A088514, the first 20 terms are the same then a(21) = 56 while A088514(21) = 25.
A229858 gives all the possible values of the smallest side a, in increasing order without repetition, but for all triples, not necessarily primitive.
All terms of A106505 are values taken by the smallest side a, in increasing order without repetition for primitive triples, but not all the lengths of this side a are present; example: 3 is not in A106505 (see comment in A229849).

			a(2) = a(5) = 7 because 2nd and 5th triple are respectively (7, 8, 13) and (7, 33, 37).

Cf. A357274 (triples), this sequence (smallest side), A357276 (middle side), A357277 (largest side), A357278 (perimeter).

Cf. also A002324, A050931, A088514, A106505, A229849.

for c from 5 to 181 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a, b)=1 and a

Formula

a(n) = A357274(n, 1).

A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

5, 8, 16, 24, 33, 35, 39, 56, 45, 63, 51, 85, 80, 57, 77, 95, 120, 120, 88, 91, 115, 143, 112, 161, 105, 175, 165, 195, 208, 160, 168, 145, 224, 203, 187, 221, 155, 261, 217, 192, 279, 209, 288, 247, 320, 272, 323, 280, 231, 315, 273, 259, 385, 357, 333, 304, 399, 352, 253, 407, 299, 287, 440

Offset: 1

Bernard Schott, Sep 25 2022

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the b's.
For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions b of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, b is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with b = 2*u*v + v^2.
This sequence is not increasing. For example, a(8) = 56 for triangle with largest side c = 61  while a(9) = 45 for triangle with largest side c = 67.
Differs from A088586, the first 20 terms are the same then a(21) = 115 while A088586(21) = 143.
A229849 gives all the possible values of the middle side b, in increasing order without repetition, for primitive triples, while A229859 gives all the possible values of the middle side b, in increasing order without repetition, but for all triples, not necessarily primitive.

			a(17) = a(18) = 120 since 17th and 18th triples are respectively (13, 120, 127) and (23, 120, 133).

Cf. A357274 (triples), A357275 (smallest side), this sequence (middle side), A357277 (largest side), A357278 (perimeter).

Cf. also A088586, A229849, A229859.

for c from 5 to 500 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a

A350347 Consider primitive 120-degree integer triangles with sides A < B < C = A002476(n). This sequence gives the values of B.

Original entry on oeis.org

5, 8, 16, 24, 33, 35, 56, 45, 63, 51, 57, 77, 95, 120, 91, 115, 143, 112, 105, 175, 165, 195, 168, 145, 224, 261, 217, 192, 288, 247, 320, 272, 280, 315, 273, 259, 385, 304, 399, 407, 299, 287, 440, 437, 301, 387, 425, 533, 416, 368, 575, 520, 423, 459, 616, 517, 441, 400, 539, 616, 637, 600, 480, 520, 728, 735, 725

Offset: 1

Seiichi Manyama, Dec 26 2021

nonn

			  n | ( A,  B,  C)
----+-------------
  1 | ( 3,  5,  7)
  2 | ( 7,  8, 13)
  3 | ( 5, 16, 19)
  4 | (11, 24, 31)
  5 | ( 7, 33, 37)
  6 | (13, 35, 43)
  7 | ( 9, 56, 61)
  8 | (32, 45, 67)
  9 | (17, 63, 73)

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Wikipedia, Integer triangle

Cf. A002365, A002476 (C), A229849, A264827, A349772 (A).

require 'prime'
def A(n)
  (1..n).each{|a|
    (a + 1..n).each{|b|
      return b if a * a + a * b + b * b == n * n
    }
  }
end
def A350347(n)
  ary = []
  i = 1
  while ary.size < n
    ary << A(i) if i.prime? && i % 6 == 1
    i += 1
  end
  ary
end
p A350347(100)

Let A = A349772(n). A^2 + A*B + B^2 = C^2.

cp's OEIS Frontend

A229858 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of A.

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A264827 (a,b,c) in lexicographic order such that a^2 + b^2 + a*b - c^2 = 0 with a < b < c and gcd(a, b) = 1.

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A229859 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of B.

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A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

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Maple

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A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.

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Maple

A350347 Consider primitive 120-degree integer triangles with sides A < B < C = A002476(n). This sequence gives the values of B.

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