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A272459
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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.
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1
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(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
(Magma) [(n*(-1-3*(-1)^n-12*n+10*n^2))/24 : n in [1..60]]; // Wesley Ivan Hurt, Sep 12 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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