A228564: Combinatorial interpretation via the natural numbers triangle

[Jason Ausborn]

Hello SeqFans,

I would like to propose a potentially new combinatorial interpretation for sequence A228564 ("Largest odd divisor of n^2 + 1").

For n >= 1, a(n) appears to be the sum of the one or two central elements of the n-th row in a triangular arrangement of the natural numbers (A000027), where row k contains k consecutive integers.

• If n is odd, a(n) is the single central element of the n-th row.

• If n is even, a(n) is the sum of the two central elements of the n-th row.

Example n=4: Row 4 is [7, 8, 9, 10]. The central elements are 8 and 9; 8 + 9 = 17, which matches a(4).

Example n=5: Row 5 is [11, 12, 13, 14, 15]. The central element is 13, which matches a(5).

This provides a unified geometric origin for the sequence's quasi-polynomial behavior and links it to the row properties of the natural numbers triangle. I have validated this for n=1 to 200.

Best regards,

Jason Ausborn

[Gareth McCaughan]

Well, it's certainly _true_. Row n of your triangle begins with n(n-1)/2+1 and ends with n(n+1)/2. The midpoint between these is (n^2+1)/2; if that's an integer then you're taking it as is, and if not you're taking twice its value which is just n^2+1. Or, to put it differently, your middle-of-triangle-row construction is equivalent to "n^2+1 if that's odd; otherwise, half of it", and since n^2+1 is never a multiple of 4 this is equivalent to "largest odd divisor of n^2+1".

But ... I don't see what's either "combinatorial" or "geometric" about this, nor how it's any more "unified" than the definition in the original title -- it literally does a case-split between odd n and even n.

--
g

[Jason Ausborn]

Thanks for the feedback!

While the algebraic result remains a case-split, the potential unified aspect of the interpretation is that the rule, sum the central element(s) of row n, is a single continuous instruction that does not require knowing the parity of n beforehand. The parity emerges based on the row lengths once it's in algebraic form.

My geometric mention is due to the spatial symmetry of the row, and combinatorial refers to the indexed positions within the partitions of A000027. I felt that the link between the number theoretic property and the structural property of the natural number triangle provides the why for the case-split behavior that is not as easily seen from the formula alone.

JA

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