A085691 - OEIS

%I #21 Jun 01 2022 17:45:28

%S 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,

%T 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,

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

%N 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.

%C 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

%C 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

%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.

%H Andrew Howroyd, <a href="/A085691/b085691.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

%F 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.

%F 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

%e Triangle begins:

%e     1;

%e     4,  1;

%e     9,  3,  1;

%e    16,  7,  3,  1;

%e    25, 13,  6,  3,  1;

%e    36, 21, 11,  6,  3,  1;

%e    49, 31, 18, 10,  6,  3,  1;

%e    64, 43, 27, 16, 10,  6,  3, 1;

%e    81, 57, 38, 24, 15, 10,  6, 3, 1;

%e   100, 73, 51, 34, 22, 15, 10, 6, 3, 1;

%e   ...

%e From _Andrew Howroyd_, Jan 05 2020: (Start)

%e 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.

%e             o

%e            / \

%e           o---o

%e          / \ / \

%e         o---o---o

%e        / \ / \ / \

%e       o---o---o---o

%e (End)

%o (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

%o (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

%Y Row sums are A002717.

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

%Y Cf. A000217, A122432, A332442.

%K nonn,tabl,easy

%O 1,2

%A _Philippe Deléham_, Jul 18 2003

%E Offset corrected and terms a(37) and beyond from _Andrew Howroyd_, Jan 05 2020