Clarification of the Definitions in A003509, A005991, and A155934

[Robert McKone]

Hello everyone,

I am reviewing https://oeis.org/A003509 NAME which reads:

Let k(m) denote the least integer such that every m X m (0,1)-matrix with exactly k(m) ones in each row and in each column contains a 2 X 2 submatrix without zeros. The sequence gives the index n of the first term in each string of equal entries in the {k(m)} sequence (see A155934).

The sequence reads k(2)-k(7) the least integer is {2, 3, 7, 13, 21, 31}.  However the confusion I am having to what this sequence actually means starts at k(4) = 7.  How can a 4 X 4 (0,1)-matrix have exactly 7 ones in each row and 7 ones in each column?  There are just not enough rows and columns.  Does this actually mean that the whole matrix actually only has 7 ones maybe, but if it does why not just say that?

The same confusion also applies to the sequence https://oeis.org/A005991 too, which effectively has the exact same NAME/definition.

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What I actually believe is the case, is that https://oeis.org/A155934 should have the NAME:

Let k(m) denote the least integer such that every m X m (0,1)-matrix with exactly k(m) ones in each row and in each column contains a 2 X 2 submatrix without zeros.

And that https://oeis.org/A005991 should just be:

The sequence gives the index n of the last term in each string of equal entries in the {k(m)} sequence (see A155934).

And that https://oeis.org/A003509 should just be:

The sequence gives the index n of the first term in each string of equal entries in the {k(m)} sequence (see A155934).

If this interpretation is correct, the three pages could perhaps be made clearer by adjusting their NAME or COMMENTS sections accordingly.

Could someone please confirm whether this reading is indeed what was intended?

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Thanks,

Robert McKone

[M F Hasler]

On Thursday, November 6, 2025 at 9:27:50 AM UTC-4 r.p.m...@gmail.com wrote:

I am reviewing https://oeis.org/A003509 NAME which reads:

Let k(m) denote the least integer such that every m X m (0,1)-matrix with exactly k(m) ones in each row and in each column contains a 2 X 2 submatrix without zeros. The sequence gives the index n of the first term in each string of equal entries in the {k(m)} sequence (see A155934).

The sequence reads k(2)-k(7) the least integer is {2, 3, 7, 13, 21, 31}.  However the confusion I am having to what this sequence actually means starts at k(4) = 7.  How can a 4 X 4 (0,1)-matrix have exactly 7 ones in each row and 7 ones in each column?  There are just not enough rows and columns.  Does this actually mean that the whole matrix actually only has 7 ones maybe, but if it does why not just say that?

The sequence is a(n), not k(m).

See the 2nd part of the definition :

a(4) = 7 is the index of ... in the {k(m)} sequence.

(But yes, I think it's confusing that it says "the index n...", it should better be "the index j", for example.)

-M.

[M F Hasler]

PS:  the sequence https://oeis.org/A005991 does not have the exact same NAME/definition :

it has "last term" instead of "first term".

Assuming that sequence  A155934(m+1) -  A155934(m) is either 0 or 1,

i.e., in contains all integers >= 2 in nondecreasing order,

I think a better definition is 

A003509(n) = index of first occurrence of n in A155934

A005991(n) = index of last occurrence of n in A155934

I propose this new definition as draft, and added examples.

Also, it would be nice to have examples in A155934.

-M.