A069011 Triangle with T(n,k) = n^2 + k^2.

0, 1, 2, 4, 5, 8, 9, 10, 13, 18, 16, 17, 20, 25, 32, 25, 26, 29, 34, 41, 50, 36, 37, 40, 45, 52, 61, 72, 49, 50, 53, 58, 65, 74, 85, 98, 64, 65, 68, 73, 80, 89, 100, 113, 128, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200

Offset: 0

Author

Henry Bottomley, Apr 02 2002

Keywords

• easy
• nonn
• tabl

Comments

For any i,j >=0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013

A227481(n) = number of squares in row n. - Reinhard Zumkeller, Oct 11 2013

Norm of the complex numbers n +- i*k and k +- i*n, where i denotes the imaginary unit. - Stefano Spezia, Aug 07 2025

Examples

			Triangle T(n,k) begins:
    0;
    1,   2;
    4,   5,   8;
    9,  10,  13,  18;
   16,  17,  20,  25,  32;
   25,  26,  29,  34,  41,  50;
   36,  37,  40,  45,  52,  61,  72;
   49,  50,  53,  58,  65,  74,  85,  98;
   64,  65,  68,  73,  80,  89, 100, 113, 128;
   81,  82,  85,  90,  97, 106, 117, 130, 145, 162;
  100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Crossrefs

Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.

Column k=0 gives A000290.

Main diagonal gives A001105.

Row sums give A132124.

T(2n,n) gives A033429.

Programs

Haskell

a069011 n k = a069011_tabl !! n !! k
a069011_row n = a069011_tabl !! n
a069011_tabl = map snd $ iterate f (1, [0]) where
   f (i, xs@(x:_)) = (i + 2, (x + i) : zipWith (+) xs [i + 1, i + 3 ..])
-- Reinhard Zumkeller, Oct 11 2013

Mathematica

Table[n^2 + k^2, {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Aug 07 2025 *)

Formula

T(n+1,k+1) = T(n,k) + 2*(n+k+1), k=0..n; T(n+1,0) = T(n,0) + 2*n + 1. - Reinhard Zumkeller, Oct 11 2013

G.f.: x*(1 + 2*y + 5*x^3*y^2 - x^2*y*(2 + 5*y) + x*(1 - 4*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Aug 04 2025