A389672 a(n) is the number of distinct pairs of integer-sided polygons obtainable by making a single straight cut through an nXn square.

0, 0, 1, 1, 3, 3, 4, 4, 7, 6, 7, 7, 14, 10, 11, 15, 16, 14, 15, 15, 19, 18, 19, 19, 41, 23, 24, 24, 29, 26, 35, 28, 34, 30, 31, 43, 49, 35, 36, 36, 59, 39, 42, 41, 48, 63, 46, 46, 87, 50, 51, 51, 59, 53, 54, 58, 76, 58, 59, 59, 126, 64, 65, 91, 77, 69, 70, 70, 80, 72

Offset

0,5

Comments

A polygon pair consists of either two rectangles, two non-rectangular right trapezoids, a right triangle and a non-rectangular right trapezoid, or a right triangle and a pentagon.

Links

Felix Huber, Table of n, a(n) for n = 0..10000

Formula

a(n) = A004526(n) + Sum_{k=1..n-1} A056137(k) + Sum_{j=1..A056137(n)} floor((n-A046083(i[j])+2)/2), where i[j] is the index of the j-th term in A046084 that equals n.

Example

a(8) = 7. Place a square of side length 8 in a Cartesian coordinate system, axis-aligned, with its lower-left corner at the origin. The 7 straight cuts are the segments [(0, 1), (8, 1)], [(0, 2), (8, 2)], [(0, 3), (8, 3)], [(0, 4), (8, 4)], [(0, 0), (8, 6)], [(0, 1), (8, 7)] and [(0, 3), (4, 0)]. No further straight cut yields a distinct pair of integer-sided polygons.

Maple

A389672:=proc(N) # To get the terms a(0) to a(N)
    local a, i, l, m, x, y, u, v;
    if N=0 then return 0 fi;
    l:=[];
    for x from 2 to floor(sqrt(floor(sqrt(2*N^2-2*N+1)-1))) do
        for y to x-1 while x^2+y^2<=floor(sqrt(2*N^2-2*N+1)-1) do
            if gcd(x, y)=1 and is(x-y, odd) then
                u:=min(x^2-y^2, 2*x*y);
                v:=max(x^2-y^2, 2*x*y);
                l:=[op(l), seq(i*[v, u], i=1..N/v)]
            fi
        od
    od;
    m:=sort([seq(l[i, 1], i=1..nops(l))]);
    a:=Vector(N);
    for i to N do
        a(i):=floor(i/2)+nops(select(x->x<i, m))
    od;
    for i in l do
       a(i[1]):=a(i[1])+floor((i[1]-i[2]+2)/2)
    od;
    return 0, op(convert(a, list))
end proc;
A389672(69);

Crossrefs

Cf. A004526, A046083, A046084, A056137, A389671.

Sequence in context: A240116 A380118 A007768 * A180018 A378765 A011371

Adjacent sequences: A389669 A389670 A389671 * A389673 A389674 A389675

Keyword

nonn,easy

Author

Felix Huber, Oct 14 2025

Status

approved