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

1,4

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.

Links

Table of n, a(n) for n=1..91.

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.

Example

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
...

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A067181 A321429 A145225 * A061480 A220692 A048962

Adjacent sequences: A332439 A332440 A332441 * A332443 A332444 A332445

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, Apr 06 2020

Status

approved