A226575 Ordered excesses of internal lattice point counts of scaled up primitive Pythagorean triangles (PPT's) (see comments).

4, 24, 48, 72, 160, 168, 180, 300, 448, 504, 520, 768, 784, 900, 1080, 1152, 1176, 1320, 1584, 1620, 1920, 2200, 2232, 2268, 2548, 2904, 3108, 3744, 3784, 3808, 3840, 4416, 4680, 4732, 5508, 5880, 5880, 5928, 6624, 6720, 6732, 7600, 8568, 8760, 9280, 9900

Offset: 1

Author

Frank M Jackson, Jun 12 2013

Keywords

• nonn

Comments

Every PPT with perpendicular legs a, b and hypotenuse c can be scaled up by the value of its hypotenuse to form a lattice triangle in two configurations. The first is where the scaled perpendicular legs a*c and b*c lie parallel to the coordinate axes. The second is where only the scaled hypotenuse c*c lies parallel to one coordinate axis. a(n) is the excess of internal lattice point counts of the second config. over the first and n is the ordered occurrence. There are multiple occurrences of this excess for different scaled PPT's. a(n) == 0 (mod 4).

Examples

			a(6) = 168 as the PPT (20,21,29) when scaled by 29 to (580,609,841) has a lattice point count of 176002 (config. 1) and 176170 (config. 2). Hence E = 168 and it is the 6th occurrence.
		

Links

• Stanley Rabinowitz, Oblique Pythagorean Lattice Triangles, Pi Mu Epsilon Journal, 9(1989), 26-29.
• Eric W. Weisstein, MathWorld: Pick's Theorem
• Wikipedia, Pick's theorem

Crossrefs

Cf. A225414, A226028.

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)(newlist[[m]][[1]]-newlist[[m]][[2]])(newlist[[m]][[2]]), {m, 1, Length[newlist]}]); maxterms=10; table=Sort@Flatten@Table[getlist[2p+1], {p, 1, maxterms}][[1;;maxterms]]

Formula

For config. 1 the internal lattice count I = (c^2*a*b-c*(a+b+1)+2)/2. For config. 2 the internal lattice count I = (c^2*a*b-(a+b+c^2)+2)/2. So the excess of config. 2 over 1 is E = (c-1)*(a+b-c)/2.