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A335681 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment. 5
0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 7, 6, 0, 0, 10, 14, 14, 10, 0, 0, 15, 22, 27, 22, 15, 0, 0, 21, 33, 42, 42, 33, 21, 0, 0, 28, 45, 63, 65, 63, 45, 28, 0, 0, 36, 60, 85, 98, 98, 85, 60, 36, 0, 0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0, 0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.
Also A335678 has colored illustrations for many values of m and n.
LINKS
FORMULA
It follows from the definitions that T(m,n] = A335680[m,n] - m - n. Note that there is an explicit formula for the latter sequence.
EXAMPLE
The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
0, 3, 7, 14, 22, 33, 45, 60, 76, 95, 115, 138, ...
0, 6, 14, 27, 42, 63, 85, 113, 143, 178, 215, 258, ...
0, 10, 22, 42, 65, 98, 131, 174, 220, 274, 330, 396, ...
0, 15, 33, 63, 98, 147, 196, 260, 329, 410, 493, 591, ...
0, 21, 45, 85, 131, 196, 261, 347, 439, 547, 656, 786, ...
0, 28, 60, 113, 174, 260, 347, 461, 583, 726, 870, 1042, ...
0, 36, 76, 143, 220, 329, 439, 583, 737, 918, 1099, 1316, ...
0, 45, 95, 178, 274, 410, 547, 726, 918, 1143, 1368, 1638, ...
0, 55, 115, 215, 330, 493, 656, 870, 1099, 1368, 1637, 1961, ...
...
The initial antidiagonals are:
0
0, 0
0, 1, 0
0, 3, 3, 0
0, 6, 7, 6, 0
0, 10, 14, 14, 10, 0
0, 15, 22, 27, 22, 15, 0
0, 21, 33, 42, 42, 33, 21, 0
0, 28, 45, 63, 65, 63, 45, 28, 0
0, 36, 60, 85, 98, 98, 85, 60, 36, 0
0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0
0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0
...
CROSSREFS
This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal case see A306302 and A331755.
Sequence in context: A330341 A152893 A335682 * A297978 A298629 A298461
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved

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Last modified May 24 12:09 EDT 2024. Contains 372773 sequences. (Running on oeis4.)