|
|
A225414
|
|
Ordered counts of internal lattice points within primitive Pythagorean triangles (PPT).
|
|
3
|
|
|
3, 22, 49, 69, 156, 187, 190, 295, 465, 498, 594, 777, 880, 931, 1144, 1269, 1330, 1501, 1611, 1633, 2190, 2272, 2494, 2619, 2655, 2893, 3475, 3732, 3937, 4182, 4524, 4719, 4900, 5502, 5635, 5866, 6490, 7021, 7185, 7719, 7761, 7828, 7849, 8688
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
Let x and y be integers used to generate the set of PPT's where x > y > 0, x + y is odd and GCD(x, y) = 1. Then the PPT area A = x*y(x^2-y^2) and the lattice points lying on the PPT boundary B = x^2-y^2+2x*y+1. Applying Pick's theorem gives internal lattice points I = A - B/2 + 1. Hence I = (x^2-y^2-1)*(2x*y-1)/2.
|
|
EXAMPLE
|
a(5)=156 as when x = 5 and n = 4, the PPT generated has area A = 180 and sides 9, 40, 41. Hence 156=180-(9+40+1)/2+1 and is the 5th such occurrence.
|
|
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)*(2 newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, {m, 1, Length[newlist]}]); maxterms = 60; Sort[Flatten[Table[getlist[2p+1], {p, 1, 10*maxterms}]]][[1;; maxterms]] (* corrected with suggestion from Giovanni Resta, May 07 2013 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|