A098921 Let [n] = {1,2,...,n}. Let G_n be the union of all closed line segments joining any two elements of [n] X [n] along a vertical or horizontal line, or along a line with slope +-1. Then a(n) = combined total of the number of (nondegenerate) triangles and rectangles for which all edges are subsets of G_n.

0, 9, 62, 211, 534, 1127, 2112, 3629, 5844, 8941, 13130, 18639, 25722, 34651, 45724, 59257, 75592, 95089, 118134, 145131, 176510, 212719, 254232, 301541, 355164, 415637, 483522, 559399, 643874, 737571, 841140, 955249, 1080592

Offset: 1

Author

Jerrold Grossman, Oct 17 2004

Keywords

• nonn
• easy

Comments

The vertices of these figures need not be in [n] X [n].

Links

• Matthew Coppenbarger, Problem 11060, The American Mathematical Monthly, Vol. 111, No. 1 (2004), p. 65; Little Boxes Made of Ticky-Tacky, Solution to Problem 11060, ibid., Vol. 112, No. 8 (2005), pp. 753-754.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Programs

Maple

F:= n -> trunc((11*n^4-2*n^3-5*n^2-22*n+18)/12);

Formula

F_n = (11n^4-2n^3-5n^2-22n+12)/12 for n even and F_n = (11n^4-2n^3-5n^2-22n+18)/12 for n odd. It can also be represented by the floor of the second expression for all n.

G.f.: -x^2*(x^4+8*x^2+26*x+9) / ((x-1)^5*(x+1)). [Colin Barker, Feb 18 2013]