A334347 Number of r X s rectangles such that r + s = 2n, where exactly one of r or s is a positive square.

0, 1, 2, 1, 1, 3, 3, 3, 3, 2, 4, 4, 3, 5, 5, 4, 3, 5, 6, 4, 6, 6, 6, 6, 4, 5, 7, 7, 5, 7, 7, 7, 8, 6, 8, 7, 6, 8, 8, 6, 7, 9, 9, 9, 7, 9, 9, 9, 8, 7, 10, 8, 8, 10, 10, 10, 10, 8, 10, 10, 9, 11, 11, 10, 7, 11, 11, 9, 11, 11, 11, 11, 10, 10, 12, 12, 12, 12, 12, 10

Offset

1,3

Links

Table of n, a(n) for n=1..80.

Formula

a(n) = Sum_{i=1..n-1} (1 - c(i))*c(2*n-i) + c(i)*(1 - c(2*n-i)), where c is the square characteristic (A010052).

Example

a(5) = 1; 2*5 = 10 has the rectangle 4 X 6, which has one square side (Note that we don't count the 1 X 9 rectangle for 10 since both sides are positive squares).
a(11) = 4; 2*11 = 22 has four rectangles with exactly one square side. They are 1 X 21, 4 X 18, 6 X 16, 9 X 13 (with squares 1, 4, 16 and 9).

Mathematica

Table[Sum[(1 - (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]])) (Floor[Sqrt[2 n - i]] - Floor[Sqrt[2 n - i - 1]]) + (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (1 - (Floor[Sqrt[2 n - i]] - Floor[Sqrt[2 n - i - 1]])), {i, n - 1}], {n, 100}]

Crossrefs

Cf. A010052.

Sequence in context: A189913 A240807 A355201 * A283672 A053268 A284828

Adjacent sequences: A334344 A334345 A334346 * A334348 A334349 A334350

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 23 2020

Status

approved