A340489 Number of distinct integer-sided convex quadrilaterals with perimeter n whose largest two sides form a right angle.

0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 2, 1, 1, 4, 2, 3, 1, 4, 5, 3, 7, 4, 6, 8, 7, 10, 6, 12, 7, 10, 16, 12, 16, 10, 18, 18, 16, 25, 18, 24, 24, 26, 30, 24, 36, 26, 34, 40, 36, 44, 34, 49, 45, 46, 58, 49, 60, 46, 64, 67, 61, 78, 64, 79, 83, 82, 91, 79, 101, 82, 99, 112, 103

Offset

0,8

Links

Table of n, a(n) for n=0..73.

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (2 - [k = j])*(-1 + sign(ceiling((k+j)/sqrt((n-i-j-k)^2 + i^2)))), where [ ] is the Iverson bracket.

Example

The notation [q,r,s,t] below shows the order in which the sides are joined (counterclockwise) starting with the largest side q, the second largest side r, and then each of the possible orders in which s and t can occur.
a(4) = 1; [1,1,1,1] a square.
a(5) = 0; ( not [2,1,1,1] since sqrt(2^2+1^2) = sqrt(5) > 1+1 = 2. )
a(7) = 2; [2,2,2,1], [2,2,1,2].
a(14) = 4; [5,3,3,3], [4,4,4,2], [4,4,3,3], and [4,4,2,4].

Mathematica

Table[Sum[Sum[Sum[(2 - KroneckerDelta[k, j]) Sign[Ceiling[(j + k)/Sqrt[(n - i - j - k)^2 + i^2]] - 1], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]

Crossrefs

Sequence in context: A361205 A355827 A139146 * A277487 A144032 A137686

Adjacent sequences: A340486 A340487 A340488 * A340490 A340491 A340492

Keyword

nonn

Author

Wesley Ivan Hurt, Jan 09 2021

Status

approved