This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354968 #78 Jul 11 2025 14:02:06
%S A354968 1,4,5,10,16,14,20,35,40,30,35,64,81,80,55,56,105,140,154,140,91,84,
%T A354968 160,220,256,260,224,140,120,231,324,390,420,405,336,204,165,320,455,
%U A354968 560,625,640,595,480,285,220,429,616,770,880,935,924,836,660,385,286,560,810,1024
%N A354968 Triangle read by rows: T(n, k) = n*k*(n+k)*(n-k)/6.
%C A354968 Given a Pythagorean triple (a,b,c), define S = c^4 - a^4 - b^4. Using Euclid's parameterization (a = 2*n*k, b = n^2 - k^2, c = n^2 + k^2), substituting to get S in terms of n and k gives S = 8*n^2*k^2*((n^2 - k^2))^2, which is a multiple of 288; T(n, k) = sqrt(S/288) = n*k*(n^2 - k^2)/6 = n*k*(n+k)*(n-k)/6.
%D A354968 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 72.
%H A354968 M. F. Hasler, <a href="/A354968/b354968.txt">Table of n, a(n) for n = 2..1000</a>, May 08 2025
%H A354968 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean Triple</a>.
%H A354968 <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples</a>.
%F A354968 G.f.: x^2*y*(1 + x*y - 4*x^2*y + x^3*y + x^4*y^2)/((1 - x)^4*(1 - x*y)^4). - _Stefano Spezia_, Jul 11 2025
%e A354968 Triangle begins:
%e A354968 n/k 1 2 3 4 5 6 7
%e A354968 2 1;
%e A354968 3 4, 5;
%e A354968 4 10, 16, 14;
%e A354968 5 20, 35, 40, 30;
%e A354968 6 35, 64, 81, 80, 55;
%e A354968 7 56, 105, 140, 154, 140, 91;
%e A354968 8 84, 160, 220, 256, 260, 224, 140;
%e A354968 ...
%e A354968 For n = 3, k = 2, a = 5, b = 12, c = 13. T(3, 2) = sqrt((13^4 - 5^4 - 12^4)/288) = 5.
%t A354968 T[n_,k_]:=n*k(n^2-k^2)/6; Table[T[n,k],{n,2,11},{k,n-1}]//Flatten (* _Stefano Spezia_, Jul 11 2025 *)
%o A354968 (PARI) apply( {A354968(n, k=0)=k|| k=n-1-(1-n=ceil(sqrt(8*n-7)/2+.5))*(2-n)\2; k*(n-k)*n*(n+k)\6}, [2..66]) \\ _M. F. Hasler_, May 08 2025
%Y A354968 Cf. A120070 (b leg), A055096 (c hypotenuse).
%Y A354968 Cf. A006414 (row sums), A000292 (column 1), A077414 (column 2), A000330 (diagonal), A107984 (transpose), A210440 (diagonal which begins with 4).
%K A354968 nonn,easy,tabl
%O A354968 2,2
%A A354968 _Ali Sada_ and _Yifan Xie_, Jun 14 2022