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A226575
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Ordered excesses of internal lattice point counts of scaled up primitive Pythagorean triangles (PPT's) (see comments).
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0
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4, 24, 48, 72, 160, 168, 180, 300, 448, 504, 520, 768, 784, 900, 1080, 1152, 1176, 1320, 1584, 1620, 1920, 2200, 2232, 2268, 2548, 2904, 3108, 3744, 3784, 3808, 3840, 4416, 4680, 4732, 5508, 5880, 5880, 5928, 6624, 6720, 6732, 7600, 8568, 8760, 9280, 9900
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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