login
Triangle read by rows: row 1 = {1}, row 2 = {1,2}. Row n consists of the smallest n distinct positive integers such that no term in row (n-1) is the average of a term in row (n-2) and a term in row n.
2

%I #8 Jun 13 2022 10:55:14

%S 1,1,2,2,4,5,1,4,5,10,1,2,7,9,10,2,5,6,7,11,12,6,16,18,19,24,25,26,2,

%T 3,4,8,9,11,12,13,9,11,13,14,15,17,19,21,22,1,2,3,4,8,12,37,43,44,45,

%U 4,6,8,12,14,16,17,18,19,20,21,2,17,18,19,42,43,44,45,46,47,48,49

%N Triangle read by rows: row 1 = {1}, row 2 = {1,2}. Row n consists of the smallest n distinct positive integers such that no term in row (n-1) is the average of a term in row (n-2) and a term in row n.

%H Rémy Sigrist, <a href="/A135018/b135018.txt">Table of n, a(n) for n = 1..10011</a>

%H Rémy Sigrist, <a href="/A135018/a135018.gp.txt">PARI program</a>

%e Triangle begins:

%e 1;

%e 1,2;

%e 2,4,5;

%e 1,4,5,10;

%e 1,2,7,9,10;

%e 2,5,6,7,11,12;

%e etc.

%e Look at row 3 (2,4,5) and row 5 (1,2,7,9,10). The averages of the terms in row 3 with the terms of row 5 are: (2+1)/2 = 1.5, (2+2)/2=2, (2+7)/2=4.5; and continuing, the averages are: 5.5, 6, (4+1)/2= 2.5, 3, 5.5, 6.5, 7, (5+1)/2=3, 3.5, 6, 7, 7.5. None of these averages occur among the terms of row 4 (1,4,5,10). In fact, the terms of row 5 are the smallest 5 distinct positive integers such that the averages with the terms of row 3 do not equal any of the terms of row 4.

%o (PARI) See Links section.

%K nonn,tabl

%O 1,3

%A _Leroy Quet_, Feb 28 2008

%E More terms from _Rémy Sigrist_, Jun 12 2022