A possible issue with A157283

[Sela Fried]

Hi, I think that the data is not consistent with the definition of the sequence 

(at least, the definition is ambiguous). 

Take, for example, n=4 and k=3. 

The corresponding matrix, as I understand it, is 

4,2,0 

5,0,0 

6,3,2 

so the determinant is -20 

but the data says -3. 

Sincerely, Sela.  

[Md. Rad Sarar Anando]

Dear Sela, 

I think you may have confusion with the term: “weakly triple-free”, which means that no subset of the form {x,2*x,3*x} with x => 1 and 3x =< n is entirely contained in the set. The definition does not involve matrices or determinants, in my opinion. 

Another point worth considering is that the entry currently provides neither a program/code for verification nor a direct formula to generate the data (perhaps due to the inherent difficulty of the problem). This entry seems somewhat confusing to me too. 

Regards,

Rad

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[Sela Fried]

Hi Rad, I don't know what you are referring to.

Please look at the following image:

‫בתאריך יום ג׳, 10 בפבר׳ 2026 ב-16:58 מאת ‪Md. Rad Sarar Anando‬‏ <‪mrad...@gmail.com‬‏>:‬

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[Md. Rad Sarar Anando]

Oh! I made a mistake by checking out  A157282.  Yeah, you were right. −20 is correct for the submatrix you chose.

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[Misha Lavrov]

The definition appears to work if we say that every determinant in the n-th row is the determinant of the k by k matrix with the "n" from the first column of A126988 in its bottom left corner. I agree that this is unintuitive.

For example, we recover the row 1,5,-10,-10,-5,1 if we take the following 6 determinants:

• A 0x0 determinant vacuously equal to 1, I guess?
• A 1x1 determinant of the matrix [5]
• A 2x2 determinant of the matrix [4,2; 5,0]
• A 3x3 determinant of the matrix [3,0,1; 4,2,0; 5,0,0]
• A 4x4 determinant of the matrix [2,1,0,0; 3,0,1,0; 4,2,0,1; 5,0,0,0]
• A 5x5 determinant of the matrix [1,0,0,0,0; 2,1,0,0,0; 3,0,1,0,0; 4,2,0,1,0; 5,0,0,0,1]

Every row of A126988 should be extended by zeroes for this definition to make sense.

Best,

Misha

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