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 A225760 Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT). 1
 2287674594, 983574906769, 16155706018465, 24267609913869, 72461523834219, 367110963344658, 473161567692022, 8504240238563547, 9271267603660839, 13796686490781630, 28200194168137420, 68964192934317607, 121927568913483970, 125247439852891719, 280877330289234924, 288885660249168850 (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 Frank A. Stevenson, Table of n, a(n) for n = 1..80 Eric Weisstein's World of Mathematics, Pick's Theorem Wikipedia, Pick's theorem FORMULA If integers a < b are the perpendicular sides of a PPT, then Pick's theorem gives the count of internal lattice points, I = (a-1)*(b-1)/2 and is comparable to the area, A = a*b/2. EXAMPLE a(1) = 2287674594 as it is the first count of internal lattice points within more than one PPT. It has (a, b) = (18108, 252685) and (28077, 162964). MATHEMATICA 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) (2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, {m, 1, Length[newlist]}]); maxterms=4000; table=Sort[Flatten[Table[getlist[2p+1], {p, 1, 2maxterms}]]]; n=1; table1={}; While[n>valuation(n, 2), a, if(gcd(b=N/a+1, a+1)==1 && issquare(b^2+(a+1)^2) && s++>1, return(1))); 0 \\ Charles R Greathouse IV, May 15 2013 CROSSREFS Cf. A024407, A225414. Sequence in context: A224988 A327056 A180688 * A345722 A346363 A022240 Adjacent sequences: A225757 A225758 A225759 * A225761 A225762 A225763 KEYWORD nonn,hard AUTHOR Frank M Jackson, May 15 2013 EXTENSIONS a(8) and beyond from Frank A. Stevenson, Nov 29 2023 STATUS approved

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Last modified June 15 03:12 EDT 2024. Contains 373402 sequences. (Running on oeis4.)