A272459 The total number of different isosceles trapezoids, excluding squares, that can be drawn on an n X n square grid where the corners of each individual trapezoid lie on a lattice point.

0, 1, 7, 18, 40, 71, 119, 180, 264, 365, 495, 646, 832, 1043, 1295, 1576, 1904, 2265, 2679, 3130, 3640, 4191, 4807, 5468, 6200, 6981, 7839, 8750, 9744, 10795, 11935, 13136, 14432, 15793, 17255, 18786, 20424, 22135, 23959, 25860, 27880, 29981, 32207, 34518

Offset: 1

Author

Christopher J. Shore, Apr 29 2016

Keywords

• nonn
• easy

Comments

This is an observation from a high school mathematics investigation: How many different isosceles trapezoids can be drawn on an n X n grid such that the corners of each individual trapezoid lie on a lattice point? The sequence gives the total number of different trapezoids that can be drawn.

There are two "families" or types of trapezoids that can be drawn on a grid. The first is where the parallel sides are drawn horizontally on the grid. The second is where the parallel sides are drawn diagonally with a gradient of 1. The number in each type follow a pattern.

1 X 1 grid: No trapezoids of either type can be drawn.

2 X 2 grid: 1 trapezoid of type 2. One parallel side is drawn diagonally through 1 square (having length sqrt(2)) and the other is drawn diagonally through two squares (length 2*sqrt(2)). Thus, the non-parallel sides are drawn horizontally or vertically to join between the parallel sides (each length 1).

3 X 3 grid: 3 trapezoids of type 1 and 4 trapezoids of type 2. The 3 trapezoids of type 1 are constructed by one parallel line drawn horizontally with length 3, the other parallel line drawn with length 1 and the perpendicular heights being successively 1, 2 and 3. Type-2 trapezoids are constructed in the same way as outlined above but with varying lengths and heights.

4 X 4 grid: 8 type-1 trapezoids and 10 type-2 trapezoids.

5 X 5 grid: 20 type-1 trapezoids and 20 type-2 trapezoids.

Hence the pattern is as follows:

Type 1 Type 2 Total

1 X 1 grid 0 0 0

2 X 2 grid 0 1 1

3 X 3 grid 3 4 7

4 X 4 grid 8 10 18

5 X 5 grid 20 20 40

6 X 6 grid 36 35 71

7 X 7 grid 63 56 119

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A000292, A032438, A143785.

Programs

Magma

[(n*(-1-3*(-1)^n-12*n+10*n^2))/24 : n in [1..60]]; // Wesley Ivan Hurt, Sep 12 2016

Maple

A272459:=n->(n*(-1-3*(-1)^n-12*n+10*n^2))/24: seq(A272459(n), n=1..60); # Wesley Ivan Hurt, Sep 12 2016

Mathematica

CoefficientList[Series[x^2 (1 + 5 x + 3 x^2 + x^3)/((1 - x)^4 (1 + x)^2), {x, 0, 44}], x] (* Michael De Vlieger, May 08 2016 *)

PARI

concat(0, Vec(x^2*(1+5*x+3*x^2+x^3)/((1-x)^4*(1+x)^2) + O(x^50))) \\ Colin Barker, May 07 2016

Formula

a(n) = Sum_{k=0..n} A032438(k) + A000292(n-1). (conjectured)

a(n) = A143785(n-2) + A000292(n-1). (conjectured)

From Colin Barker, May 07 2016: (Start)

a(n) = (n*(-1 - 3*(-1)^n - 12*n + 10*n^2))/24.

a(n) = (5*n^3 - 6*n^2 - 2*n)/12 for n even.

a(n) = (5*n^3 - 6*n^2 + n)/12 for n odd.

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.

G.f.: x^2*(1+5*x+3*x^2+x^3) / ((1-x)^4*(1+x)^2).

(End)