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 A229098 Smallest area A of Heron triangles with sides (a, b, c) in arithmetic progression of the form b - d(n), b, b + d(n), where d(n) = A091998(n) = 12*n +- 1. 1
 6, 156, 126, 546, 3750, 7350, 570, 1176, 14406, 2046, 3216, 4740, 1554, 3354, 43350, 54150, 6180, 3924, 17556, 84966, 3294, 24174, 106134, 7446, 126150, 144150, 28236, 33174, 21294, 10374, 6006, 9264, 16716, 247254, 252150, 277350, 282534, 55944, 75894, 26676 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(1) = A011945(1). According to the reference, d(n) is congruent (mod 12) to 1 or -1. Let the sides be b - d, b, b + d where 1 <= d <= b. Then the semiperimeter s = 3b/2 and by Heron's formula, the area is A = b*sqrt(3*(b^2 - 4*d^2))/4. The following table gives the first values (d(n), a, b, c, A): +------+-----+-----+-----+-------+ | d(n) |  a  |  b  |  c  |   A   | +------+-----+-----+-----+-------+ |   1  |   3 |   4 |   5 |     6 | |  11  |  15 |  26 |  37 |   156 | |  13  |  15 |  28 |  41 |   126 | |  23  |  29 |  52 |  75 |   546 | |  25  |  75 | 100 | 125 |  3750 | |  35  | 105 | 140 | 175 |  7350 | |  37  |  39 |  76 | 113 |   570 | |  47  |  51 |  98 | 145 |  1176 | |  49  | 147 | 196 | 245 | 14406 | LINKS J. A. MacDougall, Heron Triangles With Sides in Arithmetic Progression, School of Mathematical and Physical Sciences, University of Newcastle, NSW, Australia 2308, February 2, 2005. Kival Ngaokrajang, Illustration of initial terms EXAMPLE a(2) = 156 is in the sequence because d(2) = A091998(2) = 11 and (a, b, c) = (15, 26, 37) => the semiperimeter is (15 + 26 + 37)/2 = 39, and A = sqrt(39*(39-15)*(39-26)*(39-37)) = 156. MAPLE with(numtheory):u:=0:nn:=1000:lst:={1}:for k from 1 to 10 do:x:=12*k-1:y:=12*k+1:lst:=lst union {x} union {y}:od:for n from 1 to 20 do:ii:=0:d:=lst[n]:for b from 1 to nn while(ii=0)do:s:= b*sqrt(3*(b^2-4*d^2))/4:if s>0 and s=floor(s) then ii:=1:u:=u+1:printf ( "%d %d %d %d \n", u, d, b, s):else fi:od:od: CROSSREFS Cf. A011945, A091998, A188158. Sequence in context: A285368 A003460 A157626 * A203023 A128120 A237529 Adjacent sequences:  A229095 A229096 A229097 * A229099 A229100 A229101 KEYWORD nonn AUTHOR Michel Lagneau, Sep 13 2013 STATUS approved

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Last modified October 17 21:37 EDT 2019. Contains 328134 sequences. (Running on oeis4.)