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A332442
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Triangle read by rows, T(n,k) is the number of regular triangles of length k (in some length unit), for k from {1, 2, ... , n}, in a matchstick arrangement with enclosing triangle of length n, but only triangles with orientation opposite to the enclosing triangle are counted.
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1
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0, 1, 0, 3, 0, 0, 6, 1, 0, 0, 10, 3, 0, 0, 0, 15, 6, 1, 0, 0, 0, 21, 10, 3, 0, 0, 0, 0, 28, 15, 6, 1, 0, 0, 0, 0, 36, 21, 10, 3, 0, 0, 0, 0, 0, 45, 28, 15, 6, 1, 0, 0, 0, 0, 0, 55, 36, 21, 10, 3, 0, 0, 0, 0, 0, 0, 66, 45, 28, 15, 6, 1, 0, 0, 0, 0, 0, 0, 78, 55, 36, 21, 10, 3, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,4
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COMMENTS
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The matchstick arrangement consists of 3*T(n) = A000217(n) matches. One could also use a card tower with n cards as a basis.
See triangle A085691 for the number of triangles of both orientations.
See the unsigned triangle A122432 with offset 1 for the corresponding case with only the number of triangles oriented like the enclosing triangle.
The first column sequence is A000217(n-1), for n >= 1, and the following ones are then shifted downwards by 2 steps.
Row sums give A002623(n-1), n >= 1, with A002623(-1) = 0. See also the comment of Radu Grigore, Jun 19 2004, in A002623.
The nonzero terms of row 2*k are given in row k-1 of A103217, for k >= 1.
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LINKS
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FORMULA
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Recurrence: T(n, k) = T(n-1, k) + H(n-2*k+1)*(n-2*k+1), for n >=1, k = 1, 2, ..., n, and T(1, 1) = 0. Here H(x) = 1 for x >= 0 and 0 for x < 0 (a step function)..
T(n, k) = binomial(n-2*k+2, 2), for n >= 1 and k = 1, 2, ..., floor(n/2), and 0 for k = floor(n/2) + 1 .. n. See the comment by Andrew Howroyd in A085691.
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EXAMPLE
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The triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 ...
-------------------------------
1: 0
2: 1 0
3: 3 0 0
4 6 1 0 0
5: 10 3 0 0 0
6: 15 6 1 0 0 0
7: 21 10 3 0 0 0 0
8: 28 15 6 1 0 0 0 0
9: 36 21 10 3 0 0 0 0 0
10: 45 28 15 6 1 0 0 0 0 0
...
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MATHEMATICA
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T[n_, k_]:= If[k<=Floor[n/2], Binomial[n-2*k+2, 2], 0];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* Amiram Eldar, Apr 23 2020 *)
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PROG
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(PARI) T(n, k) = if(k <= n\2, binomial(n-2*k+2, 2), 0);
matrxi(10, 10, n, k, T(n, k)) \\ to see the triangle \\ Michel Marcus, May 05 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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