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

%I A225414 #19 Feb 16 2025 08:33:19
%S A225414 3,22,49,69,156,187,190,295,465,498,594,777,880,931,1144,1269,1330,
%T A225414 1501,1611,1633,2190,2272,2494,2619,2655,2893,3475,3732,3937,4182,
%U A225414 4524,4719,4900,5502,5635,5866,6490,7021,7185,7719,7761,7828,7849,8688
%N A225414 Ordered counts of internal lattice points within primitive Pythagorean triangles (PPT).
%C A225414 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.
%H A225414 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PicksTheorem.html">MathWorld: Pick's Theorem</a>
%H A225414 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pick%27s_theorem">Pick's theorem</a>
%F A225414 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.
%e A225414 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.
%t A225414 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 *)
%Y A225414 Cf. A024406.
%K A225414 nonn
%O A225414 1,1
%A A225414 _Frank M Jackson_, May 07 2013