A098921 - OEIS

%I #18 Nov 05 2025 15:35:31

%S 0,9,62,211,534,1127,2112,3629,5844,8941,13130,18639,25722,34651,

%T 45724,59257,75592,95089,118134,145131,176510,212719,254232,301541,

%U 355164,415637,483522,559399,643874,737571,841140,955249,1080592

%N 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.

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

%H Matthew Coppenbarger, <a href="https://www.jstor.org/stable/4145027">Problem 11060</a>, The American Mathematical Monthly, Vol. 111, No. 1 (2004), p. 65; <a href="https://www.jstor.org/stable/30037591">Little Boxes Made of Ticky-Tacky</a>, Solution to Problem 11060, ibid., Vol. 112, No. 8 (2005), pp. 753-754.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).

%F 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.

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

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

%K nonn,easy

%O 1,2

%A _Jerrold Grossman_, Oct 17 2004