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 A380869 #12 Mar 18 2025 10:52:19
%S A380869 8,15,20,24,27,35,39,80,84,104,143,215,220,252,264,351,363,459,476
%N A380869 Numbers k such that one can make a rectangle from a chain of linked rods of lengths 1, 2, 3, ..., k, with perimeter equal to the total length, and with one side consisting of a single rod.
%C A380869 Subsequence of A380867, with the additional requirement in the set {n1, n2, n3, n4} corresponding to the solutions (cf. there), there are two consecutive integers.
%e A380869 The smallest such number is a(1) = 8, for which we have (n1..n4) = (2, 4, 6, 7), that is, the rectangle:
%e A380869 o--+--o--o--+--+--+--+--+--+--+--o
%e A380869 | 2 1 8 |
%e A380869 |3 |
%e A380869 | |
%e A380869 o 7
%e A380869 | |
%e A380869 |4 |
%e A380869 | |
%e A380869 | 5 6 |
%e A380869 o--+--+--+--+--o--+--+--+--+--+--o
%e A380869 We see that one of the sides, of length 7, is made of only one single rod.
%o A380869 (PARI) T(n)=n*(n+1)/2 \\ = A000217
%o A380869 select( {is_A380869(n)=my(Tn=T(n), T1, T2, T3, T4, n3, n4); 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 && vecmin([n1-n2,n2-n3,n3-n4])==1 && return(n), E, )))}, [1..99])
%Y A380869 Cf. A000217 (triangular numbers), A334720 (2D cycles on square lattice).
%Y A380869 Cf. A380867 (contains this as a subsequence), A380868 (total number of solutions for given n).
%K A380869 nonn,more
%O A380869 1,1
%A A380869 _Ali Sada_ and _M. F. Hasler_, Mar 14 2025