A070216 Triangle T(n, k) = n^2 + k^2, 1 <= k <= n, read by rows.

2, 5, 8, 10, 13, 18, 17, 20, 25, 32, 26, 29, 34, 41, 50, 37, 40, 45, 52, 61, 72, 50, 53, 58, 65, 74, 85, 98, 65, 68, 73, 80, 89, 100, 113, 128, 82, 85, 90, 97, 106, 117, 130, 145, 162, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200, 122, 125, 130, 137, 146, 157

Offset: 1

Author

Charles Northup (cnorthup(AT)esc6.net), May 07 2002

Keywords

• easy
• nonn
• tabl

Comments

The formula yields squares of hypotenuses of right triangles having integer side lengths (A000404), but with duplicates (cf. A024508) and not in increasing order. - M. F. Hasler, Apr 05 2016

Examples

			a(3,2)=13 because 3^2+2^2=13.
Triangle begins:
2;
5, 8;
10, 13, 18;
17, 20, 25, 32;
26, 29, 34, 41, 50;
37, 40, 45, 52, 61, 72;
50, 53, 58, 65, 74, 85, 98;
65, 68, 73, 80, 89, 100, 113, 128;
82, 85, 90, 97, 106, 117, 130, 145, 162;
101, 104, 109, 116, 125, 136, 149, 164, 181, 200; ...
- _Vincenzo Librandi_, Apr 30 2014
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Crossrefs

Not a permutation of sequence A000404 (which has no duplicates).

Cf. A002522 (left edge), A001105 (right edge), A219054 (row sums).

Programs

Haskell

a070216 n k = a070216_tabl !! (n-1) !! (k-1)
a070216_row n = a070216_tabl !! (n-1)
a070216_tabl = zipWith (zipWith (\u v -> (u + v) `div` 2))
                       a215630_tabl a215631_tabl
-- Reinhard Zumkeller, Nov 11 2012

Magma

[n^2+k^2: k in [1..n], n in [1..15]]; // Vincenzo Librandi, Apr 30 2014

Mathematica

t[n_,k_]:=n^2 + k^2; Table[t[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Apr 30 2014 *)

PARI

T(n, k) = n^2+k^2;
for (n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ Altug Alkan, Mar 24 2016

Python

from math import isqrt
def A070216(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    return (a*(a*(a*(a-2)-(m:=n<<2)+5)+m)>>2)+n**2 # Chai Wah Wu, Jun 20 2025

Formula

a(n, k) = n^2 + k^2, 1 <= k <= n.

T(n,k) = (A215630(n,k) + A215631(n,k)) / 2, 1 <= k <=n. - Reinhard Zumkeller, Nov 11 2012

T(n,k) = A002024(n,k)^2 + A002260(n,k)^2. - David Rabahy, Mar 24 2016

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2002

Edited and corrected by M. F. Hasler, Apr 05 2016