

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
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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



FORMULA

If integers a < b are the perpendicular sides of a PPT, then Pick's theorem gives the count of internal lattice points, I = (a1)*(b1)/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]]^2newlist[[m]][[2]]^21) (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


PROG

(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


CROSSREFS



KEYWORD

nonn,hard


AUTHOR



EXTENSIONS



STATUS

approved



