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 A380868 #24 Mar 22 2025 18:40:39
%S A380868 0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,3,0,0,0,1,0,0,3,0,0,0,0,1,0,0,
%T A380868 6,0,0,0,6,0,0,0,0,6,0,0,0,1,0,0,3,0,0,0,3,3,0,0,0,0,0,0,1,0,0,0,0,3,
%U A380868 0,0,0,0,0,0,6,0,0,0,0,6,0,0,0,10,0,0,3,0,0,0,0,3,0,0,1,0,0,0,15
%N A380868 Number of distinct solutions {n1, n2, n3, n4} to the problem of forming a rectangle with sides made of linked rods of length 1, ..., n.
%C A380868 A solution to the problem is given by n1 > n2 > n3 > n4 >= 0 such that T(n) = 2*(A+B) with side lengths A = T(n1) - T(n2) = T(n3) - T(n4) and B = T(n2) - T(n3), where T = A000217.
%C A380868 When n4 = 0, this means that the rods of length 1 and n meet in a corner. When there are two consecutive n_k's (e.g. n1 = 7, n2 = 6 for n = 8) this means that one side is made of one single rod, of length equal to the larger of the two n_k's
%C A380868 Nonzero terms are at indices listed in A380867.
%C A380868 From _Daniel Mondot_, Mar 17 2025: (Start)
%C A380868 Conjecture 1: All terms are triangular numbers.
%C A380868 Conjecture 2: All triangular numbers will eventually appear in this sequence.(End)
%H A380868 Daniel Mondot, <a href="/A380868/b380868.txt">Table of n, a(n) for n = 1..10000</a>
%H A380868 Daniel Mondot, in reply to Ali Sada, <a href="https://groups.google.com/g/seqfan/c/AfihPXqYrsY/m/e0IzdVaTAgAJ">Re: A chain problem</a>, SeqFan list, Mar 13, 2025
%H A380868 Daniel Mondot, <a href="/A380868/a380868.txt">First instances of a new number of solutions</a>
%e A380868 The only solution for n = 8 is depicted in A380867.
%e A380868 a(n = 20) = 3 is the first case where we have several distinct solutions,
%e A380868 {5, 11, 15, 18}, {0, 11, 14, 18} and {0, 5, 14, 15}, The corresponding triangular numbers, whose differences give the side lengths, are {15, 66, 120, 171}, {0, 66, 105, 171} and {0, 15, 105, 120}.
%o A380868 (PARI) T(n)=n*(n+1)/2 \\ = A000217
%o A380868 apply( {A380868(n)=my(Tn=T(n), T1, T2, T3, T4, n3, n4, cnt=0); Tn%2==0 && forstep(n1=n-1, 3, -1, T1=T(n1); forstep(n2=n1-1, 2, -1, (B = Tn/2 - A = T1 - T2 = T(n2)) < 3 && break; iferr((1+n3=sqrtint(2*T3 = T2-B))*n3==2*T3 && (1+n4=sqrtint(2*T4 = T3-A))*n4==2*T4 && cnt++, E, ))); cnt}, [1..99])
%Y A380868 Cf. A000217 (triangular numbers), A334720 (2D cycles on square lattice).
%Y A380868 Cf. A380867 (indices of nonzero terms).
%K A380868 nonn
%O A380868 1,20
%A A380868 _M. F. Hasler_, Mar 14 2025