cp's OEIS Frontend

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.

%I A272459 #40 Sep 08 2022 08:46:16
%S A272459 0,1,7,18,40,71,119,180,264,365,495,646,832,1043,1295,1576,1904,2265,
%T A272459 2679,3130,3640,4191,4807,5468,6200,6981,7839,8750,9744,10795,11935,
%U A272459 13136,14432,15793,17255,18786,20424,22135,23959,25860,27880,29981,32207,34518
%N 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.
%C A272459 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.
%C A272459 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.
%C A272459 1 X 1 grid: No trapezoids of either type can be drawn.
%C A272459 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).
%C A272459 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.
%C A272459 4 X 4 grid: 8 type-1 trapezoids and 10 type-2 trapezoids.
%C A272459 5 X 5 grid: 20 type-1 trapezoids and 20 type-2 trapezoids.
%C A272459 Hence the pattern is as follows:
%C A272459               Type 1    Type 2    Total
%C A272459 1 X 1 grid       0         0        0
%C A272459 2 X 2 grid       0         1        1
%C A272459 3 X 3 grid       3         4        7
%C A272459 4 X 4 grid       8        10       18
%C A272459 5 X 5 grid      20        20       40
%C A272459 6 X 6 grid      36        35       71
%C A272459 7 X 7 grid      63        56      119
%H A272459 Colin Barker, <a href="/A272459/b272459.txt">Table of n, a(n) for n = 1..1000</a>
%H A272459 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A272459 a(n) = Sum_{k=0..n} A032438(k) + A000292(n-1). (conjectured)
%F A272459 a(n) = A143785(n-2) + A000292(n-1). (conjectured)
%F A272459 From _Colin Barker_, May 07 2016: (Start)
%F A272459 a(n) = (n*(-1 - 3*(-1)^n - 12*n + 10*n^2))/24.
%F A272459 a(n) = (5*n^3 - 6*n^2 - 2*n)/12 for n even.
%F A272459 a(n) = (5*n^3 - 6*n^2 + n)/12 for n odd.
%F A272459 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.
%F A272459 G.f.: x^2*(1+5*x+3*x^2+x^3) / ((1-x)^4*(1+x)^2).
%F A272459 (End)
%p A272459 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
%t A272459 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 *)
%o A272459 (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
%o A272459 (Magma) [(n*(-1-3*(-1)^n-12*n+10*n^2))/24 : n in [1..60]]; // _Wesley Ivan Hurt_, Sep 12 2016
%Y A272459 Cf. A000292, A032438, A143785.
%K A272459 nonn,easy
%O A272459 1,3
%A A272459 _Christopher J. Shore_, Apr 29 2016