Mian-Chowla sequence

The Mian-Chowla sequence is a $B_2$ sequence with $a_1=1$ and $a_n$ for $n>2$ being the smallest integer such that each pairwise sum $a_i+a_j$ is distinct, where and likewise for $j$, that is, $1\le i\le j\le n$. The case $i=j$ is always considered.

At the beginning, with $a_1$, there is only one pairwise sum, 2. $a_2$ can be 2 since the pairwise sums then are 2, 3 and 4. $a_3$ can’t be 3 because then there would be the pairwise sums 1 + 3 = 2 + 2 = 4. Thus $a_3=4$. The sequence, listed in A005282 of Sloane’s OEIS, continues 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, … If we define $a_1=0$, the resulting sequence is the same except each term is one less.

Rachel Lewis noticed that

$$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{a_i}\equiv 2.1585$$

, a constant listed in Finch’s book.

One way to calculate the Mian-Chowla sequence in Mathematica is thus:

a = Table[1, {40}];
n = 2;
test = 1;
While[n < 41,
      mcFlag = False;
      While[Not[mcFlag],
            test++;
            a[[n]] = test;
            pairSums = Flatten[Table[a[[i]] + a[[j]], {i, n}, {j, i, n}]];
            mcFlag = TrueQ[Length[pairSums] == Length[Union[pairSums]]]
      ];
      n++
];
a

Title	Mian-Chowla sequence
Canonical name	MianChowlaSequence
Date of creation	2013-03-22 16:27:49
Last modified on	2013-03-22 16:27:49
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	6
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11B13