A332442 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.

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

Offset: 1

Author

Wolfdieter Lang, Apr 06 2020

Keywords

• nonn
• tabl
• easy

Comments

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.

Examples

			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
...
		

Crossrefs

Cf. A000217, A002623 (row sums), A085691, A103217, A122432.

Programs

Mathematica

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 *)

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

Formula

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.