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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.

%I A229098 #21 Sep 16 2017 03:44:01
%S A229098 6,156,126,546,3750,7350,570,1176,14406,2046,3216,4740,1554,3354,
%T A229098 43350,54150,6180,3924,17556,84966,3294,24174,106134,7446,126150,
%U A229098 144150,28236,33174,21294,10374,6006,9264,16716,247254,252150,277350,282534,55944,75894,26676
%N 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.
%C A229098 a(1) = A011945(1).
%C A229098 According to the reference, d(n) is congruent (mod 12) to 1 or -1.
%C A229098 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.
%C A229098 The following table gives the first values (d(n), a, b, c, A):
%C A229098 +------+-----+-----+-----+-------+
%C A229098 | d(n) |  a  |  b  |  c  |   A   |
%C A229098 +------+-----+-----+-----+-------+
%C A229098 |   1  |   3 |   4 |   5 |     6 |
%C A229098 |  11  |  15 |  26 |  37 |   156 |
%C A229098 |  13  |  15 |  28 |  41 |   126 |
%C A229098 |  23  |  29 |  52 |  75 |   546 |
%C A229098 |  25  |  75 | 100 | 125 |  3750 |
%C A229098 |  35  | 105 | 140 | 175 |  7350 |
%C A229098 |  37  |  39 |  76 | 113 |   570 |
%C A229098 |  47  |  51 |  98 | 145 |  1176 |
%C A229098 |  49  | 147 | 196 | 245 | 14406 |
%H A229098 J. A. MacDougall, <a href="https://docserver.carma.newcastle.edu.au/id/eprint/814">Heron Triangles With Sides in Arithmetic Progression</a>, School of Mathematical and Physical Sciences, University of Newcastle, NSW, Australia 2308, February 2, 2005.
%H A229098 Kival Ngaokrajang, <a href="/A229098/a229098.pdf">Illustration of initial terms</a>
%e A229098 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.
%p A229098 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:
%Y A229098 Cf. A011945, A091998, A188158.
%K A229098 nonn
%O A229098 1,1
%A A229098 _Michel Lagneau_, Sep 13 2013