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

%I A225760 #34 Feb 16 2025 08:33:19
%S A225760 2287674594,983574906769,16155706018465,24267609913869,72461523834219,
%T A225760 367110963344658,473161567692022,8504240238563547,9271267603660839,
%U A225760 13796686490781630,28200194168137420,68964192934317607,121927568913483970,125247439852891719,280877330289234924,288885660249168850
%N A225760 Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT).
%C A225760 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 A225760 Frank A. Stevenson, <a href="/A225760/b225760.txt">Table of n, a(n) for n = 1..80</a>
%H A225760 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PicksTheorem.html">Pick's Theorem</a>
%H A225760 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pick%27s_theorem">Pick's theorem</a>
%F A225760 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 A225760 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 A225760 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 A225760 (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 A225760 Cf. A024407, A225414.
%K A225760 nonn,hard
%O A225760 1,1
%A A225760 _Frank M Jackson_, May 15 2013
%E A225760 a(8) and beyond from _Frank A. Stevenson_, Nov 29 2023