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 A379744 #16 Jan 14 2025 10:43:42
%S A379744 10,5568,5302303,5279762116,5277410421368,5277177914347752,
%T A379744 5277147974562930196,5277145259376056385184,5277145005746992952994327
%N A379744 Number of primitive Pythagorean quintuples (a, b, c, d, e) with 0 < a <= b <= c <= d <= e <= 10^n.
%C A379744 A Pythagorean quintuple (x,y,z,w,u) is a solution to x^2+y^2+z^2+w^2=u^2.
%H A379744 Math Stack Exchange, <a href="https://math.stackexchange.com/questions/4580041/pythagorean-quintuple-x2y2z2w2-u2-has-infinite-solutions-is-it-diffic">Pythagorean quintuple x^2+y^2+z^2+w^2=u^2 has infinite solutions. Is it difficult to find the general formula of its integer solution?</a>.
%F A379744 Limit_{n -> oo} a(n)/ 10^(3*n) = 5/(96*Pi^2) ~ 0.005277144981371758929368722042173314526269...
%F A379744 a(n) ~ 5*10^(3*n)/(96*Pi^2) + (3/A - 1/G)*10^(2*n)/64 + (1/(2*sqrt(3)) - 1/(4*sqrt(2)))*10^n/Pi, where A is the Dirichlet L-function value evaluated at s = 2 for the Dirichlet character with modulus 8 and index 4, and G is the Catalan's constant. (A ~ 1.064734171043503370392827451461668889483, G ~ 0.9159655941772190150546035149323841107741)
%e A379744 a(1) = 10 because there are ten primitive solutions (a, b, c, d, e) as follows: (1, 1, 1, 1, 2), (1, 1, 3, 5, 6), (1, 1, 7, 7, 10), (1, 2, 2, 4, 5), (1, 3, 3, 9, 10), (1, 4, 4, 4, 7), (1, 5, 5, 7, 10), (2, 2, 3, 8, 9), (2, 2, 4, 5, 7), and (2, 4, 5, 6, 9) with e <= 10.
%Y A379744 Cf. A101931, A157085, A225207.
%K A379744 nonn,more
%O A379744 1,1
%A A379744 _Asif Ahmed_, Dec 31 2024