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 A225414 Ordered counts of internal lattice points within primitive Pythagorean triangles (PPT). 3
 3, 22, 49, 69, 156, 187, 190, 295, 465, 498, 594, 777, 880, 931, 1144, 1269, 1330, 1501, 1611, 1633, 2190, 2272, 2494, 2619, 2655, 2893, 3475, 3732, 3937, 4182, 4524, 4719, 4900, 5502, 5635, 5866, 6490, 7021, 7185, 7719, 7761, 7828, 7849, 8688 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Eric W. Weisstein, MathWorld: Pick's Theorem Wikipedia, Pick's theorem FORMULA 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. EXAMPLE 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. MATHEMATICA getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[], #[]]==1 &]]; getlist[j_] := (newlist=getpairs[j]; Table[(newlist[[m]][]^2-newlist[[m]][]^2-1)*(2 newlist[[m]][]*newlist[[m]][]-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 *) CROSSREFS Cf. A024406. Sequence in context: A079039 A209987 A041103 * A187694 A104604 A139272 Adjacent sequences:  A225411 A225412 A225413 * A225415 A225416 A225417 KEYWORD nonn AUTHOR Frank M Jackson, May 07 2013 STATUS approved

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Last modified April 6 08:44 EDT 2020. Contains 333268 sequences. (Running on oeis4.)