A046790 -id:A046790 - cp's OEIS frontend

A308222 Numbers that are the perimeter of a primitive Heronian isosceles triangle.

16, 18, 36, 50, 64, 98, 100, 144, 162, 196, 242, 256, 324, 338, 400, 450, 484, 576, 578, 676, 722, 784, 882, 900, 1024, 1058, 1156, 1250, 1296, 1444, 1458, 1600, 1682, 1764, 1922, 1936, 2116, 2178, 2304, 2450, 2500, 2704, 2738, 2916, 3042, 3136

Offset: 1

Peter Kagey, May 16 2019

nonn

A primitive Heronian triangle is a triangle with integer sides and area, where the side lengths do not share a common divisor.

			Table illustrating first six terms of the sequence:
  perimeter |   sides    | area
  ----------+------------+-----
      16    |  (5,5,6)   |  12
      18    |  (5,5,8)   |  12
      36    | (10,13,13) |  60
      50    | (13,13,24) |  60
      50    | (16,17,17) | 120
      64    | (14,25,25) | 168
      64    | (17,17,30) | 120
      98    | (24,37,37) | 420
      98    | (25,25,48) | 168
      98    | (29,29,40) | 420

Mathematics Stack Exchange user "George", Semiperimeter of isosceles Heronian triangles.
Wikipedia, Isosceles Heronian triangles.

Cf. A046790, A096033, A096468.

a(n) = 2*A096033(n+2) (conjectured).

A308828 Number of sequences that include all residues modulo n starting with x_0 = 0 and then x_i given recursively by x_{i+1} = a * x_i + c (mod n) where a and c are integers in the interval [0..n-1].

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 8, 18, 4, 10, 4, 12, 6, 8, 32, 16, 18, 18, 8, 12, 10, 22, 16, 100, 12, 162, 12, 28, 8, 30, 128, 20, 16, 24, 36, 36, 18, 24, 32, 40, 12, 42, 20, 72, 22, 46, 64, 294, 100, 32, 24, 52, 162, 40, 48, 36, 28, 58, 16, 60, 30, 108, 512, 48, 20, 66, 32, 44, 24

Offset: 1

Haris Ziko, Jun 27 2019

nonn

All such sequences will be cyclic with period length n.
Different values of the integers a and c modulo n will always produce different sequences.
Only sequences starting with x_0 = 0 are included here. However using any other fixed number in the interval [0..n-1] for x_0 will result in the same cycles.
It appears that the set of distinct terms of this sequence gives A049225.
From Andrew Howroyd, Aug 21 2019: (Start)
Knuth calls these sequences linear congruential sequences. Theorem A in section 3.2.1.2 of the reference states that the period has length n if and only if:
     1) c is relatively prime to n;
     2) a - 1 is a multiple of p for every prime p dividing n;
     3) a - 1 is a multiple of 4 if n is a multiple of 4.
(End)

			For n = 1:
  a = 0, c = 0: [0];
  #cycles = 1 -> a(1) = 1.
For n = 5:
  a = 1, c = 1: [0, 1, 2, 3, 4];
  a = 1, c = 2: [0, 2, 4, 1, 3];
  a = 1, c = 3: [0, 3, 1, 4, 2];
  a = 1, c = 4: [0, 4, 3, 2, 1];
  #cycles = 4 -> a(5) = 4.
For n = 8:
  a = 1, c = 1: [0, 1, 2, 3, 4, 5, 6, 7];
  a = 1, c = 3: [0, 3, 6, 1, 4, 7, 2, 5];
  a = 1, c = 5: [0, 5, 2, 7, 4, 1, 6, 3];
  a = 1, c = 7: [0, 7, 6, 5, 4, 3, 2, 1];
  a = 5, c = 1: [0, 1, 6, 7, 4, 5, 2, 3];
  a = 5, c = 3: [0, 3, 2, 5, 4, 7, 6, 1];
  a = 5, c = 5: [0, 5, 6, 3, 4, 1, 2, 7];
  a = 5, c = 7: [0, 7, 2, 1, 4, 3, 6, 5];
  #cycles = 8 -> a(8) = 8.

D. E. Knuth, The Art of Computer Programming, Vol. 3, Random Numbers, Section 3.2.1.2, p. 16.

Andrew Howroyd, Table of n, a(n) for n = 1..500 (terms 1..100 from Jinyuan Wang)

Cf. A000010(phi), A026741, A003557, A046790, A049225, A135616.

checkFullSet(v,n)={my(v2=vector(n), unique=1); for(i=1, n, my(j=v[i]+1); if(v2[j]==1, unique=0; break, v2[j]=1;);); unique;};
doCycle(a,c,m)={my(v_=vector(m), x=c); v_[1]=c; for(i=1, m-1, v_[i+1]=(a*v_[i]+c)%m;); v_;};
getCycles(m)={my(L=List()); for(a=0, m-1, for(c=0, m-1, my(v1=doCycle(a,c,m)); if(checkFullSet(v1,m), listput(L, v1)););); Mat(Col(L))};
a(n)={my(M=getCycles(n)); matsize(M)[1]};

Conjecture: a(n) = A135616(n)/n.

Conjecture: a(n) = phi(n)*A003557(n / gcd(n,2)). - Andrew Howroyd, Aug 20 2019

A309395 Number of integer-sided triangles with sides a,b,c, max(a,b) < c, a + c = n that are right triangles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 3, 4, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 6, 1, 4

Offset: 1

Seiichi Manyama, Jul 28 2019

nonn

			   n | (a,b,c)
-----+-----------------------------------------
   8 | [ 3,  4,  5]
   9 | [ 4,  3,  5]
  16 | [ 6,  8, 10]
  18 | [ 5, 12, 13], [ 8,  6, 10]
  24 | [ 9, 12, 15]
  25 | [ 8, 15, 17], [12,  5, 13]
  27 | [12,  9, 15]
  32 | [ 7, 24, 25], [12, 16, 20], [15, 8, 17]
  36 | [10, 24, 26], [16, 12, 20]

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

Cf. A004526, A024155, A046790, A082523.

{a(n) = sum(k=n\2+1, n-1, issquare(k^2-(n-k)^2))}

a(n) > 0 if and only if n is a term in A046790.

a(n^2) = floor((n-1)/2) = A004526(n-1).

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A308222 Numbers that are the perimeter of a primitive Heronian isosceles triangle.

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A308828 Number of sequences that include all residues modulo n starting with x_0 = 0 and then x_i given recursively by x_{i+1} = a * x_i + c (mod n) where a and c are integers in the interval [0..n-1].

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A309395 Number of integer-sided triangles with sides a,b,c, max(a,b) < c, a + c = n that are right triangles.

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