A225414 Ordered counts of internal lattice points within primitive Pythagorean triangles (PPT).

3, 22, 49, 69, 156, 187, 190, 295, 465, 498, 594, 777, 880, 931, 1144, 1269, 1330, 1501, 1611, 1633, 2190, 2272, 2494, 2619, 2655, 2893, 3475, 3732, 3937, 4182, 4524, 4719, 4900, 5502, 5635, 5866, 6490, 7021, 7185, 7719, 7761, 7828, 7849, 8688

Offset: 1

Author

Frank M Jackson, May 07 2013

Keywords

• nonn

Comments

A PPT can be drawn as a closed lattice polygon with the hypotenuse intersecting no lattice points other than at its start and end. Consequently the PPT is subject to Pick's theorem.

Examples

			a(5)=156 as when x = 5 and n = 4, the PPT generated has area A = 180 and sides 9, 40, 41. Hence 156=180-(9+40+1)/2+1 and is the 5th such occurrence.
		

Links

• Eric W. Weisstein, MathWorld: Pick's Theorem
• Wikipedia, Pick's theorem

Crossrefs

Cf. A024406.

Programs

Mathematica

getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getlist[j_] := (newlist=getpairs[j]; Table[(newlist[[m]][[1]]^2-newlist[[m]][[2]]^2-1)*(2 newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, {m, 1, Length[newlist]}]); maxterms = 60; Sort[Flatten[Table[getlist[2p+1], {p, 1, 10*maxterms}]]][[1;;maxterms]] (* corrected with suggestion from Giovanni Resta, May 07 2013 *)

Formula

Let x and y be integers used to generate the set of PPT's where x > y > 0, x + y is odd and GCD(x, y) = 1. Then the PPT area A = x*y(x^2-y^2) and the lattice points lying on the PPT boundary B = x^2-y^2+2x*y+1. Applying Pick's theorem gives internal lattice points I = A - B/2 + 1. Hence I = (x^2-y^2-1)*(2x*y-1)/2.