A085691 Triangle read by rows: T(n,k) is the number of triangles of side k in triangular matchstick arrangement of side n; n>=1 and k>=1.

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, 121, 91, 66, 46, 31, 21, 15, 10, 6, 3, 1, 144, 111, 83, 60, 42, 29, 21, 15, 10, 6, 3, 1

Offset: 1

Author

Philippe Deléham, Jul 18 2003

Keywords

• nonn
• tabl
• easy

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

Examples

			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)
		

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)

Crossrefs

Row sums are A002717.

Columns k=1..3 are A000290, A002061, A010000.

Cf. A000217, A122432, A332442.

Programs

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

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

Extensions

Offset corrected and terms a(37) and beyond from Andrew Howroyd, Jan 05 2020