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A226028
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Array T(j,k) of counts of internal lattice points within all Pythagorean triangles (see comments for array order).
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1
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3, 22, 17, 49, 103, 43, 69, 217, 244, 81, 156, 305, 505, 445, 131, 187, 671, 709, 913, 706, 193, 190, 793, 1546, 1281, 1441, 1027, 267, 295, 799, 1819, 2781, 2021, 2089, 1408, 353, 465, 1249, 1828, 3265, 4376, 2929, 2857, 1849, 451, 498, 1937, 2863, 3277, 5131, 6331, 4005, 3745, 2350, 561
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OFFSET
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1,1
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COMMENTS
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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.
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.
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LINKS
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EXAMPLE
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Array begins
3, 17, 43, 81, 131, ...
22, 103, 244, 445, ...
49, 217, 505, ...
69, 305, ...
156, ...
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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