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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

Table of n, a(n) for n=1..44.

Eric W. Weisstein, MathWorld: Pick's Theorem

Wikipedia, Pick's theorem

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

Cf. A024406.

Sequence in context: A079039 A209987 A041103 * A187694 A104604 A139272

Adjacent sequences:  A225411 A225412 A225413 * A225415 A225416 A225417

KEYWORD

nonn

AUTHOR

Frank M Jackson, May 07 2013

STATUS

approved

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Last modified September 22 16:56 EDT 2019. Contains 327311 sequences. (Running on oeis4.)