OFFSET
1,2
COMMENTS
See A296305 for the corresponding sums.
Each term may be involved in up to three sums:
- T(1, 1) is involved in one sum,
- For any n > 1, T(n, 1) and T(n, k) are involved in two sums:
- For any n > 1 and k such that 1 < k < n, T(n, k) is involved in three sums.
The parity of the terms of the triangle has interesting features:
- For any n > 35:
- T(n, 1) is even,
- T(n, k) is odd for any k such that 1 < k < n - 34,
- T(n, n - 34) is even,
- T(n, n - k) and T(n + 64, n + 64 - k) have the same parity for k=0..34,
- See representation in Links section (the black pattern visible alongside the right border is eventually periodic),
- These features also appear in the scatterplot of the triangle as a flat sequence in the form of two branches: the first branch above the X=Y axis corresponds to the (frequent) odd terms, and the dashed branch under the X=Y axis corresponds to the (sparse) even terms.
- for three distinct positive numbers to sum to a prime number, either all of them are odd or two of them are even and the other is odd,
- we have both situations here,
- we have only the first situation in A073671,
- we have only the second situation in A076990.
See also A297615 for a similar triangle.
LINKS
Rémy Sigrist, Rows n = 1..100, flattened
Rémy Sigrist, Colored representation of the first 500 rows (where the color is function of the parity of T(n, k))
Rémy Sigrist, PARI program for A297673
EXAMPLE
Triangle begins:
1: 1
2: 2, 4
3: 3, 6, 7
4: 5, 9, 8, 14
5: 10, 16, 12, 11, 18
6: 13, 20, 17, 24, 26, 15
7: 19, 21, 30, 32, 23, 22, 34
8: 25, 27, 31, 28, 29, 37, 38, 35
9: 33, 39, 41, 55, 44, 36, 40, 49, 43
10: 42, 52, 46, 50, 58, 47, 48, 51, 57, 63
The term T(1, 1) = 1 is involved in the following sum:
- 1 + 2 + 4 = 7.
The term T(3, 3) = 7 is involved in the following sums:
- 4 + 6 + 7 = 17,
- 7 + 8 + 14 = 29.
The term T(4, 2) = 9 is involved in the following sums:
- 3 + 5 + 9 = 17,
- 6 + 9 + 8 = 23,
- 9 + 16 + 12 = 37.
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Jan 03 2018
STATUS
approved