OFFSET
1,2
COMMENTS
Sub-triangles can be oriented in one of two ways. The number of sub-triangles that are oriented in the same way as the full triangle is binomial(n-k+2, 2). For k <= n/2, there are also sub-triangles oriented at 180 degrees and the number of these is binomial(n-2*k+2, 2). - Andrew Howroyd, Jan 06 2020
The matchstick arrangement consists of 3*A000217(n) matchsticks. One can also consider it as a tower of cards with n base cards. - Wolfdieter Lang, Apr 06 2020
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
FORMULA
T(n, k) = 0 for n < k; T(n, k) = (n-k+1)*(n-k+2)/2 for k <= n < 2*k; T(n, k) = n^2 - 3*(k-1)*n + (5*k-4)*(k-1)/2 for 2*k <= n.
T(n, k) = Tup(n, k) + Tdown(n, k), with Tup(n, k) = (-1)*(n-k)*A122432(n-1, k-1) and Tdown(n, k) = A332442(n, k), for n >= 1, and k = 1, 2, ..., n. - Wolfdieter Lang, Apr 06 2020
EXAMPLE
Triangle begins:
1;
4, 1;
9, 3, 1;
16, 7, 3, 1;
25, 13, 6, 3, 1;
36, 21, 11, 6, 3, 1;
49, 31, 18, 10, 6, 3, 1;
64, 43, 27, 16, 10, 6, 3, 1;
81, 57, 38, 24, 15, 10, 6, 3, 1;
100, 73, 51, 34, 22, 15, 10, 6, 3, 1;
...
From Andrew Howroyd, Jan 05 2020: (Start)
Row n=3: In the triangle illustrated below there are 9 small triangles, 3 triangles with side length 2 and 1 with side length 3.
o
/ \
o---o
/ \ / \
o---o---o
/ \ / \ / \
o---o---o---o
(End)
PROG
(PARI) T(n, k)={binomial(n-k+2, 2) + if(2*k<=n, binomial(n-2*k+2, 2), 0)} \\ Andrew Howroyd, Jan 06 2020
(PARI) T(n, k)={if(k>n, 0, if(2*k > n, (n-k+1)*(n-k+2)/2, n^2 - 3*(k-1)*n + (5*k-4)*(k-1)/2))} \\ Andrew Howroyd, Jan 06 2020
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Jul 18 2003
EXTENSIONS
Offset corrected and terms a(37) and beyond from Andrew Howroyd, Jan 05 2020
STATUS
approved