

A334347


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


0



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
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OFFSET

1,3


LINKS

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


FORMULA

a(n) = Sum_{i=1..n1} (1  c(i))*c(2*ni) + c(i)*(1  c(2*ni)), 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: A320902 A189913 A240807 * A283672 A053268 A284828
Adjacent sequences: A334344 A334345 A334346 * A334348 A334349 A334350


KEYWORD

nonn,easy


AUTHOR

Wesley Ivan Hurt, Apr 23 2020


STATUS

approved



