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

2287674594, 983574906769, 16155706018465, 24267609913869, 72461523834219, 367110963344658, 473161567692022, 8504240238563547, 9271267603660839, 13796686490781630, 28200194168137420, 68964192934317607, 121927568913483970, 125247439852891719, 280877330289234924, 288885660249168850

Offset: 1

Author

Frank M Jackson, May 15 2013

Keywords

• nonn
• hard

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.

Examples

			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).
		

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

Crossrefs

Cf. A024407, A225414.

Programs

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
				

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

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.

Extensions

a(8) and beyond from Frank A. Stevenson, Nov 29 2023