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 A382268 #20 Apr 03 2025 13:22:46
%S A382268 15,20,24,35,39,44,48,55,56,63,75,76,80,84,91,95,99,104,111,119,120,
%T A382268 132,135,140,143,144,152,155,168,175,176,187,188,195,203,207,215,216,
%U A382268 219,224,252,259,260,264,272,275,279,287,288,296,299,308,315,320,324,335,351,360
%N A382268 Numbers k such that a right triangle can be formed from a chain of linked rods of lengths 1, 2, 3, ..., k, with the perimeter equal to the total length.
%C A382268 The corresponding perimeters T(a(n)) must be in the intersection of T = A000217 (triangular numbers) with A010814 (perimeters of integer sided right triangles). A number k is in the sequence if there exists a solution {k1, k2, k3} with k > k1 > k2 > k3 >= 0 such that for a < b < c in { T(k1) - T(k2), T(k2) - T(k3), T(k) - T(k1) + T(k3) } one has a^2 + b^2 = c^2. - _M. F. Hasler_, Mar 20 2025
%H A382268 Daniel Mondot, <a href="/A382268/b382268.txt">Table of n, a(n) for n = 1..3052</a>
%e A382268 The first triangle is when k = 15. The segments are [6+7+8+9+10] [11+12+13+14] [15+1+2+3+4+5]. The sums of the segments are (40, 50, 30), which is 10 times the primitive right triangle (3, 4, 5).
%e A382268 The second term, k = 20, corresponds to 5 distinct solutions:
%e A382268 S1 = {18, 16, 9}: a = 9+...+1 + 20+19 = 84, b = 18+17 = 35, c = 16+...+10 = 91,
%e A382268 S2 = {17, 11, 3}: a = 20+19+18 + 3+2+1 = 63, c = 17+...+12 = 87, b = 11+...+4 = 60,
%e A382268 S3 = {17, 11, 2}: a = 20+19+18 + 2+1 = 60, c = 17+...+12 = 87, b = 11+...+3 = 63,
%e A382268 S4 = {16, 9, 4}: a = 20+...+17 + 4+...+1 = 84, c = 16+...+10 = 91, b = 9+...+5 = 35,
%e A382268 S5 = {15, 8, 1}: c = 20+...+16 + 1 = 91, a = 15+...+9 = 84, b = 8+...+2 = 35.
%e A382268 We note that S2 and S3, and S1, S4 and S5, have the same side lengths, but different decompositions.
%o A382268 (PARI) select( {is_A382268(n)=my(Tn=n*(n+1)\2,T1,T2,S); Tn%2==0 && is_A005279(Tn\2) && forstep(n1=n-1,sqrtint(Tn-1)+1,-1, T1=n1*(n1+1)\2; forstep(n2=n1-1,sqrtint(2*T1-Tn-1)+1,-1, T2=n2*(n2+1)\2; forstep(n3=n2-1,0,-1, #(S=Set([Tn-T1+S=n3*(n3+1)\2,T2-S,T1-T2]))>2 && S[3]^2 == S[1]^2+S[2]^2 && return(S))))}, [1..100])\\ _M. F. Hasler_, Mar 22 2025
%Y A382268 Cf. A000217, A010814, A380867, A380868, A380875.
%K A382268 nonn
%O A382268 1,1
%A A382268 _Ali Sada_ and _Daniel Mondot_, Mar 19 2025