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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.
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%I #20 Dec 21 2024 02:35:17

%S 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,

%T 21,33,42,42,33,21,0,0,28,45,63,65,63,45,28,0,0,36,60,85,98,98,85,60,

%U 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

%N 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.

%C The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

%C Also A335678 has colored illustrations for many values of m and n.

%H <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>

%F 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.

%e The initial rows of the array are:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...

%e 0, 3, 7, 14, 22, 33, 45, 60, 76, 95, 115, 138, ...

%e 0, 6, 14, 27, 42, 63, 85, 113, 143, 178, 215, 258, ...

%e 0, 10, 22, 42, 65, 98, 131, 174, 220, 274, 330, 396, ...

%e 0, 15, 33, 63, 98, 147, 196, 260, 329, 410, 493, 591, ...

%e 0, 21, 45, 85, 131, 196, 261, 347, 439, 547, 656, 786, ...

%e 0, 28, 60, 113, 174, 260, 347, 461, 583, 726, 870, 1042, ...

%e 0, 36, 76, 143, 220, 329, 439, 583, 737, 918, 1099, 1316, ...

%e 0, 45, 95, 178, 274, 410, 547, 726, 918, 1143, 1368, 1638, ...

%e 0, 55, 115, 215, 330, 493, 656, 870, 1099, 1368, 1637, 1961, ...

%e ...

%e The initial antidiagonals are:

%e 0

%e 0, 0

%e 0, 1, 0

%e 0, 3, 3, 0

%e 0, 6, 7, 6, 0

%e 0, 10, 14, 14, 10, 0

%e 0, 15, 22, 27, 22, 15, 0

%e 0, 21, 33, 42, 42, 33, 21, 0

%e 0, 28, 45, 63, 65, 63, 45, 28, 0

%e 0, 36, 60, 85, 98, 98, 85, 60, 36, 0

%e 0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0

%e 0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0

%e ...

%Y This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.

%Y For the diagonal case see A306302 and A331755.

%K nonn,tabl

%O 1,8

%A _Lars Blomberg_, _Scott R. Shannon_, and _N. J. A. Sloane_, Jun 28 2020