Floyd’s Triangle Diagonals & Combinatorial Sequences

Overview

Floyd’s Triangle is a simple arrangement of the natural numbers in a triangular “ramp”:

1
2   3
4   5   6
7   8   9  10
11 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54 55

By grouping numbers along diagonals we generate two interesting families of sums:

Slope‑Right Diagonals (negative‑slope): For group k (k ≥ 1), using rows r = k, k+1, …, 2k–1, we select the element in position L(r) = 2·(r – k) + 1. Their sums yield the structured tetragonal anti‑prism numbers with the formula:
S_right(k) = (7k³ – 3k² + 2k)/6. ( OEIS: A100182)
Slope‑Left Diagonals (mirror selection): For group k, from the same rows (r = k to 2k–1) we select the element in position R(r) = 2k – r. Their sums follow:
S_left(k) = (7k³ – 6k² + 5k)/6. ( OEIS Search: 1, 7, 25, 62, 125, 221, …)

With a sufficiently large Floyd’s Triangle (at least 2k–1 rows for group k), these selections yield complete cubic sequences.

Visualization: Slope‑Right Diagonals (Flush Left)

1
2   3
4   5   6
7   8   9  10
11  12  13  14  15
16  17  18  19  20  21
22  23  24  25  26  27  28
29  30  31  32  33  34  35  36
37  38  39  40  41  42  43  44  45
46  47  48  49  50  51  52  53  54  55

Visualization: Slope‑Left Diagonals (Flush Right)

          1
       2   3
    4   5   6
 7   8   9   10
11  12  13  14  15
16  17  18  19  20  21
22  23  24  25  26  27  28
29  30  31  32  33  34  35  36
37  38  39  40  41  42  43  44  45
46  47  48  49  50  51  52  53  54  55

Verification: Diagonal Sums

Slope‑Right Diagonal Sums

For group k (using rows r = k to 2k–1), the slope‑right diagonal sum is:
S_right(k) = (7k³ – 3k² + 2k)/6.

Slope‑Right Diagonal Sums
Group k	Formula	Value
1	(7×1³ – 3×1² + 2×1)/6	1
2	(7×8 – 3×4 + 2×2)/6	8
3	(7×27 – 3×9 + 2×3)/6	28
4	(7×64 – 3×16 + 2×4)/6	68
5	(7×125 – 3×25 + 2×5)/6	135
6	(7×216 – 3×36 + 2×6)/6	236
7	(7×343 – 3×49 + 2×7)/6	378
8	(7×512 – 3×64 + 2×8)/6	568
9	(7×729 – 3×81 + 2×9)/6	813
10	(7×1000 – 3×100 + 2×10)/6	1120

Slope‑Left Diagonal Sums

For group k, the slope‑left diagonal sum is:
S_left(k) = (7k³ – 6k² + 5k)/6.

Slope‑Left Diagonal Sums
Group k	Formula	Value
1	(7×1³ – 6×1² + 5×1)/6	1
2	(7×8 – 6×4 + 5×2)/6	7
3	(7×27 – 6×9 + 5×3)/6	25
4	(7×64 – 6×16 + 5×4)/6	62
5	(7×125 – 6×25 + 5×5)/6	125
6	(7×216 – 6×36 + 5×6)/6	221
7	(7×343 – 6×49 + 5×7)/6	357
8	(7×512 – 6×64 + 5×8)/6	540
9	(7×729 – 6×81 + 5×9)/6	777
10	(7×1000 – 6×100 + 5×10)/6	1075

Discussion & Further Resources

The slope‑right diagonal sums correspond to the structured tetragonal anti‑prism numbers (OEIS A100182), given by the formula:
S_right(k) = (7k³ – 3k² + 2k)/6.

The slope‑left diagonal sums form a related cubic sequence:
S_left(k) = (7k³ – 6k² + 5k)/6.

For additional details and related sequences, please see:
OEIS: Structured Tetragonal Anti‑Prism Numbers (Slope‑Right Diagonals)
and
OEIS Search: 1, 7, 25, 62, 125, 221, … (Slope‑Left Diagonals).

These investigations reveal that even a simple arrangement such as Floyd’s Triangle contains hidden layers of combinatorial beauty and symmetry.