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

1,2

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

Row sums are A002717.

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

Cf. A000217, A122432, A332442.

Sequence in context: A331149 A331145 A339635 * A055461 A324999 A104796

Adjacent sequences: A085688 A085689 A085690 * A085692 A085693 A085694

Keyword

nonn,tabl,easy

Author

Philippe Deléham, Jul 18 2003

Extensions

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

Status

approved