OFFSET
1,1
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).
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
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.
EXAMPLE
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.
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]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Jun 12 2013
STATUS
approved