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A240240
Consider primitive Heronian triangles with integer area and with sides {m, m+1, c}, where c > m+1. The sequence gives the possible values of m.
0
3, 9, 13, 19, 20, 33, 51, 65, 73, 99, 119, 129, 163, 170, 174, 193, 201, 203, 220, 243, 260, 269, 287, 289, 339, 362, 377, 393, 450, 451, 513, 532, 559, 579, 615, 649, 696, 702, 714, 723, 740, 771, 801, 883, 909, 940, 969, 975, 1059, 1112, 1153, 1155, 1156, 1164, 1251, 1299, 1325, 1332, 1353, 1424, 1455, 1459, 1569, 1605, 1615, 1683, 1690, 1716, 1801, 1869, 1919, 1923
OFFSET
1,1
COMMENTS
Corresponding values of c are 5, 17, 15, 37, 29, 65, 101, 109, 145.
And corresponding values of area/6 are 1, 6, 14, 19, 35, 44, 85, 330, 146, 231, 1190.
The sequence includes all terms of A016064 (where c = m+2) except for the first term, 1 (case with zero area).
Note that in all cases c is odd and m+2 <= c < 2m+1.
EXAMPLE
First triangle has sides (3,4,5) and area 6.
2nd triangle has sides (9,10,17) and area 36.
3rd triangle has sides (13,14,15) and area 84.
MATHEMATICA
re=Reap[Do[a=m; b=m+1; Do[s=(a+b+c)/2; area=Sqrt[s(s-a)(s-b)(s-c)]; If[IntegerQ[area], Sow[{a, b, c, area}]; Break[]], {c, 2m-1, m+2, -2 }], {m, 3, 2000}]][[2, 1]]; #[[1]]&/@ re
CROSSREFS
KEYWORD
nonn
AUTHOR
Zak Seidov, Apr 03 2014
STATUS
approved