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A225760 Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT). 0

%I

%S 2287674594,983574906769,16155706018465,24267609913869,72461523834219,

%T 367110963344658,473161567692022

%N Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT).

%C 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.

%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>

%F 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.

%e 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).

%t 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<Length[table], (If[table[[n+1]]==table[[n]], table1=Append[table1, table[[n]]]]; n++)]; table1

%o (PARI) is(n)=my(b,s,N=2*n);fordiv(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

%Y Cf. A024407, A225414.

%K nonn,more

%O 1,1

%A _Frank M Jackson_, May 15 2013

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Last modified April 5 10:33 EDT 2020. Contains 333239 sequences. (Running on oeis4.)