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.

8, 15, 20, 24, 27, 35, 39, 80, 84, 104, 143, 215, 220, 252, 264, 351, 363, 459, 476

Offset: 1

Author

Ali Sada and M. F. Hasler, Mar 14 2025

Keywords

• nonn
• more

Comments

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.

Examples

			The smallest such number is a(1) = 8, for which we have (n1..n4) = (2, 4, 6, 7), that is, the rectangle:
    o--+--o--o--+--+--+--+--+--+--+--o
    |  2   1             8           |
    |3                               |
    |                                |
    o                                7
    |                                |
    |4                               |
    |                                |
    |       5               6        |
    o--+--+--+--+--o--+--+--+--+--+--o
We see that one of the sides, of length 7, is made of only one single rod.
		

Crossrefs

Cf. A000217 (triangular numbers), A334720 (2D cycles on square lattice).

Cf. A380867 (contains this as a subsequence), A380868 (total number of solutions for given n).

Programs

PARI

T(n)=n*(n+1)/2 \\ = A000217
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])