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

1,1

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<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

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