A294457 Sum of all the diagonals of the distinct rectangles that can be made with positive integer sides such that L + W = n, W < L. The sums are then rounded to the nearest integer.

0, 0, 4, 6, 15, 19, 33, 38, 57, 64, 87, 97, 124, 135, 168, 181, 218, 232, 274, 291, 337, 355, 407, 427, 482, 504, 565, 589, 654, 679, 749, 777, 851, 880, 959, 991, 1074, 1107, 1196, 1230, 1323, 1360, 1458, 1496, 1599, 1639, 1746, 1788, 1900, 1944, 2060, 2106

Offset

1,3

Links

Table of n, a(n) for n=1..52.

Formula

a(n) = round(2 * Sum_{i=1..floor((n-1)/2)} sqrt(i^2 + (n-i)^2)).

Example

a(4) = 6; There is only one 1 X 3 rectangle (there is no 2 X 2 rectangle since W < L) and sqrt(1^2 + 3^2) = sqrt(10). Since there are two diagonals in a rectangle, the total length is 2*sqrt(10). Then we have round(2*sqrt(10)) = round(6.32455532..) = 6.

Mathematica

Table[Round[2*Sum[Sqrt[i^2 + (n - i)^2], {i, Floor[(n-1)/2]}]], {n, 80}]

Crossrefs

Cf. A050187.

Sequence in context: A117883 A106387 A034771 * A266883 A034764 A119034

Adjacent sequences: A294454 A294455 A294456 * A294458 A294459 A294460

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 30 2017

Status

approved