bibkey: irwin2016a138099
authors: Paul Curtz and Benedict W. J. Irwin
year: 2016
title: "OEIS A138099: irregular triangle and Irwin's hook-sum conjecture"
doi: null
url: https://oeis.org/A138099
claim: "Irwin conjectures that each term is the hook-sum increment of the displayed spiral tableaux."
strata_touched:
  - D5/S3/Combinatorics/Polyomino/IrwinA138099HookSum
license: citation-only
triage: anchor

Irwin’s hook-sum conjecture for OEIS A138099

Verified locator

Source: https://oeis.org/A138099

OEIS A138099, revision 26, NAME, FORMULA and EXAMPLE blocks (unpaginated). The sequence is attributed to Paul Curtz; the hook-sum conjecture is attributed to Benedict W. J. Irwin. The source has no DOI or arXiv identifier.

Source text

Irregular triangle read by rows: T(n,k) = k + floor(n/2), 1 <= k <= ceiling(n/2).

From _Benedict W. J. Irwin_, May 14 2016: (Start)
Conjecture: a(n) = HookSum(T(n))-HookSum(T(n-1)), where HookSum(A) is the sum of all hooks across a tableau A, and T(n) are the spiraling sequence of tableaux:
                                    o   oo  ooo
Empty, o, oo, oo, oo, oo  , ooo , ooo, ooo, ooo, ...
              o   oo  ooo   ooo   ooo  ooo  ooo
(End)

Triangle begins:
1;
2;
2, 3;
3, 4;
3, 4, 5;
4, 5, 6;
4, 5, 6, 7;
5, 6, 7, 8;
...

The source uses one-based sequence indices. The arithmetic rows have lengths 1, 1, 2, 2, 3, 3, and so on. The diagrams are interpreted as their occupied integer cells, with ordinary hooks containing the cell and cells strictly right in its row or below in its column. The square-spiral walk starts at the origin, travels west, south, east and north, and uses leg lengths 1, 1, 2, 2, 3, 3, and so on. The Lean assertion includes existence of both source sequences and states the increment identity for every positive index. Hook sums are cast to integers before subtraction.

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