A353077 - OEIS

%I #40 Oct 05 2024 13:57:22

%S 0,0,1,0,1,3,0,1,3,9,0,1,4,14,16,0,1,3,8,12,18,-1,-1,-1,-1,-1,-1,-1,0,

%T 1,3,13,32,36,43,52,0,1,3,7,15,31,36,54,63,0,1,3,9,27,49,56,61,77,81,

%U -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,3,12,20,34,38,81,88,94,104,109

%N Triangle read by rows, where the n-th row consists of the lexicographically earliest solution for n integers in 0..p-1 whose n*(n-1) differences are congruent to 1..p-1 (mod p), where p=n*(n-1)+1. If no solution exists, the n-th row consists of n -1's.

%H Martin Becker, <a href="/A353077/b353077.txt">Rows n = 1..200 of triangle, flattened</a>.

%H Leonard E. Dickson, <a href="https://doi.org/10.2307/2968498">Problem 142</a>, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), pp. 107-108.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectDifferenceSet.html">Perfect Difference Set</a>

%e    n   row

%e    1 [0];

%e    2 [0,1];

%e    3 [0,1,3];

%e    4 [0,1,3,9];

%e    5 [0,1,4,14,16];

%e    6 [0,1,3,8,12,18];

%e    7 no solution exists;

%e    8 [0,1,3,13,32,36,43,52];

%e    9 [0,1,3,7,15,31,36,54,63];

%e   10 [0,1,3,9,27,49,56,61,77,81];

%e   11 no solution exists;

%e   12 [0,1,3,12,20,34,38,81,88,94,104,109];

%e   13 no solution exists;

%e   14 [0,1,3,16,23,28,42,76,82,86,119,137,154,175];

%e   15 no solution exists;

%e   16 no solution exists.

%o (PARI) isok(n, v) = my(p=n*(n-1)+1); setbinop((x,y)->lift(Mod(x-y, p)), v, v) == [0..p-1];

%o row(n) = forsubset([n^2-n+1, n], s, my(ds = apply(x->x-1, Vec(s))); if (isok(n, ds), return(ds)););

%Y Cf. A002061, A058241, A351690, A333852.

%K sign,look,tabl

%O 1,6

%A _Michel Marcus_, Apr 22 2022

%E Name and data corrected for "lexicographically earliest solution" by _Michel Marcus_, May 09 2022

%E Adjusted to a regular triangle, and rows 1, 2, 7, and 10-12 inserted by _Pontus von Brömssen_, May 09 2022