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%I #36 Sep 08 2022 08:45:05
%S 1,2,3,6,7,10,13,16,19,24,27,32,37,42,47,54,59,66,73,80,87,96,103,112,
%T 121,130,139,150,159,170,181,192,203,216,227,240,253,266,279,294,307,
%U 322,337,352,367,384,399,416,433,450,467,486,503,522,541,560,579
%N Maximum number of triangles in polyiamond with perimeter n.
%H Colin Barker, <a href="/A069813/b069813.txt">Table of n, a(n) for n = 3..1000</a>
%H W. C. Yang, R. R. Meyer, <a href="http://digital.library.wisc.edu/1793/64366">Maximal and minimal polyiamonds</a>, 2002.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F a(n) = round(n^2/6) - (0 if n = 0 mod 6, 1 else) = A056829(n)-A097325(n).
%F From _Colin Barker_, Jan 18 2015: (Start)
%F a(n) = round((-25 + 9*(-1)^n + 8*exp(-2/3*i*n*Pi) + 8*exp((2*i*n*Pi)/3) + 6*n^2)/36), where i=sqrt(-1).
%F G.f.: x^3*(1+x-x^2)*(1+x^2) / ((1-x)^3*(1+x)*(1+x+x^2)). (End)
%F a(n) = A001399(n-3) + A001399(n-4) + A001399(n-6) - A001399(n-7). - _R. J. Mathar_, Jul 14 2015
%e a(10) = 16 because the maximum number of triangles in a polyiamond of perimeter 10 is 16.
%p A069813 := proc(n)
%p round(n^2/6) ;
%p if modp(n,6) <> 0 then
%p %-1 ;
%p else
%p % ;
%p end if;
%p end proc: # _R. J. Mathar_, Jul 14 2015
%t LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 2, 3, 6, 7, 10}, 60] (* _Jean-François Alcover_, Jan 03 2020 *)
%o (PARI) a(n) = round(n^2/6) - (n % 6 != 0) \\ _Michel Marcus_, Jul 17 2013
%o (PARI) Vec(x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^60)) \\ _Colin Barker_, Jan 19 2015
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)))); // _Marius A. Burtea_, Jan 03 2020
%Y Cf. A000105, A000577, A027709, A030511 (bisection), A057729, A067628.
%K nonn,easy
%O 3,2
%A Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002
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