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%I #11 Mar 03 2024 14:44:48
%S 8,9,16,25,40,89,144,233,520,841,1360,3033,4904,7929,17680,28585,
%T 46216,103049,166608,269369,600616,971065,1570000,3500649,5659784,
%U 9150633,20403280,32987641,53333800,118919033,192266064,310852169,693110920
%N Numbers n such that 17, n, n+1 are the sides of a Heron triangle, i.e., a triangle with integer sides and integer area.
%C Numbers n such that the square root of 72*(n-8)*(n+9) is an integer. - _Harvey P. Dale_, Jun 23 2013
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 6, -6, 0, -1, 1).
%F a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7), with a(1)=8, a(2)=9, a(3)=16, a(4)=25, a(5)=40, a(6)=89, a(7)=144
%F G.f.: (-9*x^6 - 7*x^5 - 9*x^4 + 39*x^3 - 7*x^2 - x - 8)/(x^7 - x^6 - 6*x^4 + 6*x^3 + x - 1).
%e For n=2 the triangle has sides 17,9,10 with area 36.
%p A145820 := proc (n) local L; option remember;
%p L := [8, 9, 16, 25, 40, 89, 144];
%p if n < 8 then L[n] else A145820(n-1)+6*A145820(n-3)-6*A145820(n-4)-A145820(n-6)+A145820(n-7) end if;
%p end proc;
%p seq(A145820(n),n=1..50);
%t LinearRecurrence[{1,0,6,-6,0,-1,1},{8,9,16,25,40,89,144},40] (* _Harvey P. Dale_, Jun 23 2013 *)
%Y Cf. A145821.
%K nonn
%O 1,1
%A _James R. Buddenhagen_, Oct 20 2008
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