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A242497 Sides of (Heronian) triangles where sides are consecutive integers and area is an integer.

%I A242497 #22 Sep 04 2023 12:25:52
%S A242497 3,4,5,13,14,15,51,52,53,193,194,195,723,724,725,2701,2702,2703,10083,
%T A242497 10084,10085,37633,37634,37635,140451,140452,140453,524173,524174,
%U A242497 524175,1956243,1956244,1956245,7300801,7300802,7300803,27246963,27246964,27246965
%N A242497 Sides of (Heronian) triangles where sides are consecutive integers and area is an integer.
%C A242497 Let the edge lengths of the triangle be 2x-1, 2x, 2x+1 so that area = sqrt{3x * x * (x-1) * (x+1)} and we need x^2 - 1 to be of shape 3y^2.  That is, x/y is an even rank convergent to the continued fraction of sqrt(3) and x is A001075.
%C A242497 The intermediate length sides are given by A003500(n), n >= 1. Note that A003500(0) = 2 corresponds to the degenerate (Heronian) triangle with sides {1, 2, 3} and area 0. - _Daniel Forgues_, May 28 2014
%D A242497 Nakane Genkei (Nakane the Elder), Shichijo Beki Yenshiki, 1691.
%H A242497 Harvey P. Dale, <a href="/A242497/b242497.txt">Table of n, a(n) for n = 1..1000</a>
%H A242497 David Eugene Smith and Yoshio Mikami, <a href="http://books.google.com/books?id=J1YNAAAAYAAJ&amp;pg=PA168&amp;lpg=PA168&amp;dq=%22nakane+solves+it%22&amp;source=bl&amp;ots=SXl7poRJQr&amp;sig=-PH4VG63ZAPk-YXus2EjEE5TeS0&amp;hl=en&amp;sa=X&amp;ei=yqSLUfHtA67b4AO42oC4Cg&amp;ved=0CBgQ6AEwAA#v=onepage&amp;q=%22nakane%20solves%20it%22&amp;f=false">A history of Japanese mathematics</a>, Dover, 2004, p. 168.
%H A242497 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,4,4,4,-1,-1,-1).
%F A242497 G.f.: (-3*x^7 - 5*x^6 - 6*x^5 + 4*x^4 + 10*x^3 + 12*x^2 + 7*x + 3)/ ((1+x+x^2)*(1-4*x^3+x^6)). - _R. J. Mathar_, May 30 2023
%t A242497 LinearRecurrence[{-1,-1,4,4,4,-1,-1,-1},{3,4,5,13,14,15,51,52},40] (* _Harvey P. Dale_, May 04 2021 *)
%o A242497 (PARI) Vec((-3*x^7 - 5*x^6 - 6*x^5 + 4*x^4 + 10*x^3 + 12*x^2 + 7*x + 3)/(x^8 + x^7+ x^6 - 4*x^5 - 4*x^4 - 4*x^3 + x^2 + x + 1)+O(x^99))
%Y A242497 A016064 is the main entry for this sequence.
%K A242497 nonn,easy
%O A242497 1,1
%A A242497 _R. K. Guy_ and _Charles R Greathouse IV_, May 16 2014