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A253547 Total number of star-shaped dodecagons appearing in a variant of hexagon expansion after n iterations.

%I A253547 #24 Mar 21 2025 11:15:51
%S A253547 0,0,0,1,3,9,16,23,33,43,56,69,85,101,120,139,161,183,208,233,261,289,
%T A253547 320,351,385,419,456,493,533,573,616,659,705,751,800,849,901,953,1008,
%U A253547 1063,1121,1179,1240,1301,1365,1429,1496,1563,1633,1703,1776,1849,1925,2001,2080
%N A253547 Total number of star-shaped dodecagons appearing in a variant of hexagon expansion after n iterations.
%C A253547 Total number of hexagons after n iterations is A179178. See illustration.
%H A253547 Kival Ngaokrajang, <a href="/A253547/a253547.pdf">Illustration of initial terms</a>
%H A253547 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A253547 Conjectures from _Colin Barker_, Jan 03 2015: (Start)
%F A253547 a(n) = (27 - 3*(-1)^n - 28*n + 6*n^2)/8 for n>5.
%F A253547 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>9.
%F A253547 G.f.: -x^4*(2*x^5 - 4*x^4 + 3*x^2 + x + 1) / ((x-1)^3*(x+1)). (End)
%t A253547 LinearRecurrence[{2,0,-2,1},{0,0,0,1,3,9,16,23,33},60] (* _Harvey P. Dale_, Oct 30 2015 *)
%o A253547 (PARI)
%o A253547 {
%o A253547 a=1;d1=0;print1("0, 0, 0, 1",", ");
%o A253547 for(n=4,100,
%o A253547    if(n<5,d1=2,
%o A253547      if(n<6,d1=6,
%o A253547        if(n<7,d1=7,
%o A253547          if(Mod(n,2)==0,d1=d1+3
%o A253547          )
%o A253547        )
%o A253547      )
%o A253547    );
%o A253547    a=a+d1;
%o A253547    print1(a,", ")
%o A253547 )
%o A253547 }
%Y A253547 Cf. A179178.
%K A253547 nonn,nice,easy
%O A253547 1,5
%A A253547 _Kival Ngaokrajang_, Jan 03 2015