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A226028 Array T(j,k) of counts of internal lattice points within all Pythagorean triangles (see comments for array order). 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The array of counts of internal lattice points within all Pythagorean triangles T(j,k) is arranged such 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: A122495 A100977 A037101 * A248626 A072398 A134924

Adjacent sequences:  A226025 A226026 A226027 * A226029 A226030 A226031

KEYWORD

nonn,tabl

AUTHOR

Frank M Jackson, May 23 2013

STATUS

approved

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Last modified March 20 05:45 EDT 2019. Contains 321344 sequences. (Running on oeis4.)