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%I #16 Apr 27 2019 21:00:53
%S 3,22,17,49,103,43,69,217,244,81,156,305,505,445,131,187,671,709,913,
%T 706,193,190,793,1546,1281,1441,1027,267,295,799,1819,2781,2021,2089,
%U 1408,353,465,1249,1828,3265,4376,2929,2857,1849,451,498,1937,2863,3277,5131,6331,4005,3745,2350,561
%N Array T(j,k) of counts of internal lattice points within all Pythagorean triangles (see comments for array order).
%C The array of counts of internal lattice points within all Pythagorean triangles T(j,k) is arranged so that its first column is the ordered counts of internal lattice points within the k-th primitive Pythagorean triangle (PPT) A225414(k) and the j-th column is j multiples of these PPT side lengths.
%C Let the k-th PPT have integer perpendicular sides a, b then its j-th multiple has area A = j^2*a*b/2 and the count of lattice points intersected by its boundary is B = j*(a+b+1) by the application of Pick's theorem the count of internal lattice points within it is I = (j^2*a*b-j*(a+b+1)+2)/2.
%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>
%e Array begins
%e 3, 17, 43, 81, 131, ...
%e 22, 103, 244, 445, ...
%e 49, 217, 505, ...
%e 69, 305, ...
%e 156, ...
%t getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getpptpairs[j_] := (newlist=getpairs[j]; Table[{(newlist[[m]][[1]]^2-newlist[[m]][[2]]^2-1)(2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, newlist[[m]][[1]]^2-newlist[[m]][[2]]^2, 2newlist[[m]][[1]]*newlist[[m]][[2]]}, {m, 1, Length[newlist]}]); lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@ IntegerPartitions[#1 +dim-1, {dim}], 1] &, maxHeight], 1]; array[{x_, y_}] := (pptpair=table[[y]]; (x^2*pptpair[[2]]*pptpair[[3]])/2-x(pptpair[[2]]+pptpair[[3]]+1)/2+1); maxterms=20; table=Sort[Flatten[Table[getpptpairs[2p+1], {p, 1, maxterms}], 1]][[1;;maxterms]]; pairs=lexicographicLattice[{2, maxterms}]; Table[array[pairs[[n]]], {n, 1, maxterms(maxterms+1)/2}]
%Y Cf. A126587 (first row), A225414 (first column).
%K nonn,tabl
%O 1,1
%A _Frank M Jackson_, May 23 2013
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