A226028 Array T(j,k) of counts of internal lattice points within all Pythagorean triangles (see comments for array order).

3, 22, 17, 49, 103, 43, 69, 217, 244, 81, 156, 305, 505, 445, 131, 187, 671, 709, 913, 706, 193, 190, 793, 1546, 1281, 1441, 1027, 267, 295, 799, 1819, 2781, 2021, 2089, 1408, 353, 465, 1249, 1828, 3265, 4376, 2929, 2857, 1849, 451, 498, 1937, 2863, 3277, 5131, 6331, 4005, 3745, 2350, 561

Offset

1,1

Comments

The array of counts of internal lattice points within all Pythagorean triangles T(j,k) is arranged so that its first column is the ordered counts of internal lattice points within the k-th primitive Pythagorean triangle (PPT) A225414(k) and the j-th column is j multiples of these PPT side lengths.

Let the k-th PPT have integer perpendicular sides a, b then its j-th multiple has area A = j^2*a*b/2 and the count of lattice points intersected by its boundary is B = j*(a+b+1) by the application of Pick's theorem the count of internal lattice points within it is I = (j^2*a*b-j*(a+b+1)+2)/2.

Links

Table of n, a(n) for n=1..55.

Eric W. Weisstein, MathWorld: Pick's Theorem

Wikipedia, Pick's theorem

Example

Array begins
    3,   17,  43,  81, 131, ...
   22,  103, 244, 445, ...
   49,  217, 505, ...
   69,  305, ...
  156, ...

Mathematica

getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getpptpairs[j_] := (newlist=getpairs[j]; Table[{(newlist[[m]][[1]]^2-newlist[[m]][[2]]^2-1)(2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, newlist[[m]][[1]]^2-newlist[[m]][[2]]^2, 2newlist[[m]][[1]]*newlist[[m]][[2]]}, {m, 1, Length[newlist]}]); lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@ IntegerPartitions[#1 +dim-1, {dim}], 1] &, maxHeight], 1]; array[{x_, y_}] := (pptpair=table[[y]]; (x^2*pptpair[[2]]*pptpair[[3]])/2-x(pptpair[[2]]+pptpair[[3]]+1)/2+1); maxterms=20; table=Sort[Flatten[Table[getpptpairs[2p+1], {p, 1, maxterms}], 1]][[1;; maxterms]]; pairs=lexicographicLattice[{2, maxterms}]; Table[array[pairs[[n]]], {n, 1, maxterms(maxterms+1)/2}]

Crossrefs

Cf. A126587 (first row), A225414 (first column).

Sequence in context: A100977 A364163 A037101 * A248626 A360366 A072398

Adjacent sequences: A226025 A226026 A226027 * A226029 A226030 A226031

Keyword

nonn,tabl

Author

Frank M Jackson, May 23 2013

Status

approved