A055096 Triangle read by rows, sums of 2 distinct nonzero squares: T(n,k) = k^2+n^2, (n>=2, 1 <= k <= n-1)

5, 10, 13, 17, 20, 25, 26, 29, 34, 41, 37, 40, 45, 52, 61, 50, 53, 58, 65, 74, 85, 65, 68, 73, 80, 89, 100, 113, 82, 85, 90, 97, 106, 117, 130, 145, 101, 104, 109, 116, 125, 136, 149, 164, 181, 122, 125, 130, 137, 146, 157, 170, 185, 202, 221, 145, 148, 153, 160

Offset

2,1

Comments

Discovered by Bernard Frénicle de Bessy (1605?-1675). - Paul Curtz, Aug 18 2008

Terms that are not hypotenuses in primitive Pythagorean triangles, are replaced by 0 in A222946. - Reinhard Zumkeller, Mar 23 2013

This triangle T(n,k) gives the circumdiameters for the Pythagorean triangles with a = (n+1)^2 - k^2, b = 2*(n+1)*k and c = (n+1)^2 + k^2 (see the Floor van Lamoen entries or comments A063929, A063930, A002283, A003991). See also the formula section. Note that not all Pythagorean triangles are covered, e.g., (9,12,15) does not appear. - Wolfdieter Lang, Dec 03 2014

Links

Reinhard Zumkeller, Rows n = 2..121 of triangle, flattened

M. de Frénicle, Méthode pour trouver la solutions des problèmes par les exclusions, in: "Divers ouvrages de mathématiques et de physique, par Messieurs de l'Académie royale des sciences", Paris, 1693, pp 1-44.

Antti Karttunen, Larger table, showing also locations of 4k+1 primes and squares

Eric Weisstein's World of Mathematics, Congruum Problem.

Index entries for sequences related to sums of squares

Formula

a(n) = sum2distinct_squares_array(n).

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

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

T(2*n-2, n-1) = A033429(n-1).

T(n,k) = A133819(n,k) + A140978(n,k) = (n+1)^2 + k^2, 1 <= k <= n. - Reinhard Zumkeller, Mar 23 2013

T(n, k) = a*b*c/(2*sqrt(s*(s-1)*(s-b)*(s-c))) with s =(a + b + c)/2 and the substitution a = (n+1)^2 - k^2, b = 2*(n+1)*k and c = (n+1)^2 + k^2 (the circumdiameter for the considered Pythagorean triangles). - Wolfdieter Lang, Dec 03 2014

From Bob Selcoe, Mar 21 2015: (Start)

T(n,k) = 1 + (n-k+1)^2 + Sum_{j=0..k-2} (4*j + 2*(n-k+3)).

T(n,k) = 1 + (n+k-1)^2 - Sum_{j=0..k-2} (2*(n+k-3) - 4*j).

Therefore: 4*(n-k+1) + Sum_{j=0..k-2} (2*(n-k+3) + 4*j) = 4*n(k-1) - Sum_{j=0..k-2} (2*(n+k-3) - 4*j). (End)

From G. C. Greubel, Apr 19 2023: (Start)

T(2*n-3, n-1) = A033429(n-1).

T(2*n-4, n-2) = A079273(n-1).

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

T(3*n-4, n-1) = 10*A000290(n-1) = A033583(n-1).

Sum_{k=1..n-1} T(n, k) = A331987(n-1).

Sum_{k=1..floor(n/2)} T(n-k, k) = A226141(n-1). (End)

Example

The triangle T(n, k) begins:
n\k   1   2   3   4   5   6   7   8   9  10  11 ...
2:    5
3:   10  13
4:   17  20  25
5:   26  29  34  41
6:   37  40  45  52  61
7:   50  53  58  65  74  85
8:   65  68  73  80  89 100 113
9:   82  85  90  97 106 117 130 145
10: 101 104 109 116 125 136 149 164 181
11: 122 125 130 137 146 157 170 185 202 221
12: 145 148 153 160 169 180 193 208 225 244 265
...
13: 170 173 178 185 194 205 218 233 250 269 290 313,
14: 197 200 205 212 221 232 245 260 277 296 317 340 365,
15: 226 229 234 241 250 261 274 289 306 325 346 369 394 421,
16: 257 260 265 272 281 292 305 320 337 356 377 400 425 452 481,
...
Formatted and extended by Wolfdieter Lang, Dec 02 2014 (reformatted Jun 11 2015)
The successive terms are (1^2+2^2), (1^2+3^2), (2^2+3^2), (1^2+4^2), (2^2+4^2), (3^2+4^2), ...

Maple

sum2distinct_squares_array := (n) -> (((n-((trinv(n-1)*(trinv(n-1)-1))/2))^2)+((trinv(n-1)+1)^2));

Mathematica

T[n_, k_]:= (n+1)^2 + k^2; Table[T[n, k], {n, 15}, {k, n}]//Flatten (* Jean-François Alcover, Mar 16 2015, after Reinhard Zumkeller *)

Prog

(Haskell)
a055096 n k = a055096_tabl !! (n-1) !! (k-1)
a055096_row n = a055096_tabl !! (n-1)
a055096_tabl = zipWith (zipWith (+)) a133819_tabl a140978_tabl
-- Reinhard Zumkeller, Mar 23 2013
(Magma) [n^2+k^2: k in [1..n-1], n in [2..15]]; // G. C. Greubel, Apr 19 2023
(SageMath)
def A055096(n, k): return n^2 + k^2
flatten([[A055096(n, k) for k in range(1, n)] for n in range(2, 16)]) # G. C. Greubel, Apr 19 2023

Crossrefs

Sorting gives A024507. Count of divisors: A055097, Möbius: A055132. For trinv, follow A055088.

Cf. A001844 (right edge), A002522 (left edge), A033429 (central column).

Cf. A000290, A033583, A079273, A190816, A226141, A331987.

Sequence in context: A025302 A268379 A221265 * A313386 A267969 A132777

Adjacent sequences: A055093 A055094 A055095 * A055097 A055098 A055099

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 04 2000

Extensions

Edited: in T(n, k) formula by Reinhard Zumkeller k < n replaced by k <= n. - Wolfdieter Lang, Dec 02 2014

Made definition more precise, changed offset to 2. - N. J. A. Sloane, Mar 30 2015

Status

approved