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%I #11 Jun 14 2013 15:02:14
%S 4,24,48,72,160,168,180,300,448,504,520,768,784,900,1080,1152,1176,
%T 1320,1584,1620,1920,2200,2232,2268,2548,2904,3108,3744,3784,3808,
%U 3840,4416,4680,4732,5508,5880,5880,5928,6624,6720,6732,7600,8568,8760,9280,9900
%N Ordered excesses of internal lattice point counts of scaled up primitive Pythagorean triangles (PPT's) (see comments).
%C 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).
%H Stanley Rabinowitz, <a href="http://www.mathpropress.com/stan/bibliography/oblique.pdf">Oblique Pythagorean Lattice Triangles</a>, Pi Mu Epsilon Journal, 9(1989), 26-29.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PicksTheorem.html">MathWorld: Pick's Theorem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pick%27s_theorem">Pick's theorem</a>
%F 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.
%e 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.
%t 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]]
%Y Cf. A225414, A226028.
%K nonn
%O 1,1
%A _Frank M Jackson_, Jun 12 2013
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