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A225760 Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT). 1
2287674594, 983574906769, 16155706018465, 24267609913869, 72461523834219, 367110963344658, 473161567692022, 8504240238563547, 9271267603660839, 13796686490781630, 28200194168137420, 68964192934317607, 121927568913483970, 125247439852891719, 280877330289234924, 288885660249168850 (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
Frank A. Stevenson, Table of n, a(n) for n = 1..80
Eric Weisstein's World of Mathematics, Pick's Theorem
Wikipedia, Pick's theorem
FORMULA
If integers a < b are the perpendicular sides of a PPT, then Pick's theorem gives the count of internal lattice points, I = (a-1)*(b-1)/2 and is comparable to the area, A = a*b/2.
EXAMPLE
a(1) = 2287674594 as it is the first count of internal lattice points within more than one PPT. It has (a, b) = (18108, 252685) and (28077, 162964).
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) (2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, {m, 1, Length[newlist]}]); maxterms=4000; table=Sort[Flatten[Table[getlist[2p+1], {p, 1, 2maxterms}]]]; n=1; table1={}; While[n<Length[table], (If[table[[n+1]]==table[[n]], table1=Append[table1, table[[n]]]]; n++)]; table1
PROG
(PARI) is(n)=my(b, s, N=2*n); fordiv(n>>valuation(n, 2), a, if(gcd(b=N/a+1, a+1)==1 && issquare(b^2+(a+1)^2) && s++>1, return(1))); 0 \\ Charles R Greathouse IV, May 15 2013
CROSSREFS
Sequence in context: A224988 A327056 A180688 * A345722 A346363 A022240
KEYWORD
nonn,hard
AUTHOR
Frank M Jackson, May 15 2013
EXTENSIONS
a(8) and beyond from Frank A. Stevenson, Nov 29 2023
STATUS
approved

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Last modified June 15 03:12 EDT 2024. Contains 373402 sequences. (Running on oeis4.)