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A226575 Ordered excesses of internal lattice point counts of scaled up primitive Pythagorean triangles (PPT's) (see comments). 0
4, 24, 48, 72, 160, 168, 180, 300, 448, 504, 520, 768, 784, 900, 1080, 1152, 1176, 1320, 1584, 1620, 1920, 2200, 2232, 2268, 2548, 2904, 3108, 3744, 3784, 3808, 3840, 4416, 4680, 4732, 5508, 5880, 5880, 5928, 6624, 6720, 6732, 7600, 8568, 8760, 9280, 9900 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Every PPT with perpendicular legs a, b and hypotenuse c can be scaled up by the value of its hypotenuse to form a lattice triangle in two configurations. The first is where the scaled perpendicular legs a*c and b*c lie parallel to the coordinate axes. The second is where only the scaled hypotenuse c*c lies parallel to one coordinate axis. a(n) is the excess of internal lattice point counts of the second config. over the first and n is the ordered occurrence. There are multiple occurrences of this excess for different scaled PPT's. a(n) == 0 (mod 4).

LINKS

Table of n, a(n) for n=1..46.

Stanley Rabinowitz, Oblique Pythagorean Lattice Triangles, Pi Mu Epsilon Journal, 9(1989), 26-29.

Eric W. Weisstein, MathWorld: Pick's Theorem

Wikipedia, Pick's theorem

FORMULA

For config. 1 the internal lattice count I = (c^2*a*b-c*(a+b+1)+2)/2. For config. 2 the internal lattice count I = (c^2*a*b-(a+b+c^2)+2)/2. So the excess of config. 2 over 1 is E = (c-1)*(a+b-c)/2.

EXAMPLE

a(6) = 168 as the PPT (20,21,29) when scaled by 29 to (580,609,841) has a lattice point count of 176002 (config. 1) and 176170 (config. 2). Hence E = 168 and it is the 6th 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)(newlist[[m]][[1]]-newlist[[m]][[2]])(newlist[[m]][[2]]), {m, 1, Length[newlist]}]); maxterms=10; table=Sort@Flatten@Table[getlist[2p+1], {p, 1, maxterms}][[1;; maxterms]]

CROSSREFS

Cf. A225414, A226028.

Sequence in context: A080380 A039935 A090821 * A052645 A353250 A191778

Adjacent sequences:  A226572 A226573 A226574 * A226576 A226577 A226578

KEYWORD

nonn

AUTHOR

Frank M Jackson, Jun 12 2013

STATUS

approved

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Last modified June 30 03:34 EDT 2022. Contains 354913 sequences. (Running on oeis4.)