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%I #19 Sep 08 2022 08:45:10
%S 1,1,3,6,8,12,15,19,25,30,36,42,49,55,63,72,80,90,99,109,121,132,144,
%T 156,169,181,195,210,224,240,255,271,289,306,324,342,361,379,399,420,
%U 440,462,483,505,529,552,576,600,625,649,675,702,728
%N Independence number of king KG_2 on triangle board B_n.
%H Vincenzo Librandi, <a href="/A082873/b082873.txt">Table of n, a(n) for n = 1..1000</a>
%H J.-J. Bode, H. Harborth and M. Harborth, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00825-7">King independence on triangle boards</a>, Discr. Math., 266 (2003), 101-107.
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).
%F a(n) = floor((n+1)^2/4) if n == 0, 1, 4, 6, 9, 10, 11 (mod 12), a(n) = floor((n+1)^2/4) - 1 otherwise.
%F a(n)= +2*a(n-1) -a(n-2) +a(n-12) -2*a(n-13) +a(n-14). - _R. J. Mathar_, Aug 05 2014
%F G.f.: -x*(1-x+2*x^2-x^4+2*x^5-x^6+x^7+2*x^8-x^9+x^10+x^3) / ( (1+x) *(1+x^2) *(x^4-x^2+1) *(x^2-x+1) *(1+x+x^2) *(x-1)^3 ). - _R. J. Mathar_, Aug 05 2014
%t CoefficientList[Series[-(1 - x + 2 x^2 - x^4 + 2 x^5 - x^6 + x^7 + 2 x^8 - x^9 + x^10 + x^3)/((1 + x) (1 + x^2) (x^4 - x^2 + 1) (x^2 - x + 1) (1 + x + x^2) (x - 1)^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 06 2014 *)
%o (Magma) I:=[1,1,3,6,8,12,15,19,25,30,36,42,49,55]; [n le 14 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-12)-2*Self(n-13)+Self(n-14): n in [1..60]]; // _Vincenzo Librandi_, Aug 06 2014
%Y Cf. A082874.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, May 25 2003
%E More terms from _Sean A. Irvine_, Sep 26 2011
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