A088307 Triangle, read by rows, T(n,k) = n^2 + k^2 if gcd(n,k)=1, otherwise 0.

2, 5, 0, 10, 13, 0, 17, 0, 25, 0, 26, 29, 34, 41, 0, 37, 0, 0, 0, 61, 0, 50, 53, 58, 65, 74, 85, 0, 65, 0, 73, 0, 89, 0, 113, 0, 82, 85, 0, 97, 106, 0, 130, 145, 0, 101, 0, 109, 0, 0, 0, 149, 0, 181, 0, 122, 125, 130, 137, 146, 157, 170, 185, 202, 221, 0

Offset: 1

Author

Reinhard Zumkeller, Nov 05 2003

Keywords

• nonn
• tabl

Comments

(n^2-k^2, 2*k*n, T(n,k)) is a primitive Pythagorean triple iff T(n,k) > 0.

Examples

			Triangle begins:
   2;
   5,  0;
  10, 13,  0;
  17,  0, 25,  0;
  26, 29, 34, 41,  0;
  37,  0,  0,  0, 61, 0;
  ...
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A000007, A000010, A001844, A002522, A008527, A053818.

Cf. A057427, A070216, A078370, A079273, A190816.

Programs

Magma

function A088307(n,k)
  if GCD(k,n) eq 1 then return n^2+k^2;
  else return 0;
  end if; return A088307;
end function;
[A088307(n,k): k in [1..n], n in [1..13]]; // G. C. Greubel, Dec 16 2022

Mathematica

Table[If[CoprimeQ[n,k],n^2+k^2,0],{n,20},{k,n}]//Flatten (* Harvey P. Dale, Jul 13 2018 *)

SageMath

def A088307(n,k):
    if (gcd(n,k)==1): return n^2 + k^2
    else: return 0
flatten([[A088307(n,k) for k in range(1,n+1)] for n in range(1,14)]) # G. C. Greubel, Dec 16 2022

Formula

T(n, n) = 2*A000007(n-1).

T(n, 1) = A002522(n).

T(2*n+1, 2) = A078370(n).

Sum_{k=1..n} A057427(T(n,m)) = A000010(n).

From G. C. Greubel, Dec 15 2022: (Start)

T(n, n-1) = A001844(n).

T(n, n-2) = ((1-(-1)^n)/2) * A008527((n+1)/2).

T(2*n, n) = 5*A000007(n-1).

T(2*n+1, n) = A079273(n+1).

T(2*n-1, n) = A190816(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A053818(n+1) + [n=1]. (End)