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A253688 The total number of pentagons in a variant of pentagon expansion (vertex-to-vertex, two consecutive vertices and one isolated vertex) after n iterations.

%I A253688 #7 Jan 15 2015 12:38:39
%S A253688 1,4,10,21,39,64,94,129,171,218,272,331,397,468,546,629,719,814,916,
%T A253688 1023,1137,1256,1382,1513,1651,1794,1944,2099,2261,2428,2602,2781,
%U A253688 2967,3158,3356,3559,3769,3984,4206,4433,4667,4906,5152,5403,5661,5924,6194,6469,6751,7038
%N A253688 The total number of pentagons in a variant of pentagon expansion (vertex-to-vertex, two consecutive vertices and one isolated vertex) after n iterations.
%C A253688 Two star shaped icosagons appearing at n >= 6. See illustration.
%H A253688 Kival Ngaokrajang, <a href="/A253688/a253688.pdf">Illustration of initial terms</a>
%F A253688 Conjectures from _Colin Barker_, Jan 09 2015: (Start)
%F A253688 a(n) = (53-(-1)^n-38*n+12*n^2)/4 for n>5.
%F A253688 a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>9.
%F A253688 G.f.: -x*(2*x^8-2*x^7-2*x^6+2*x^5+4*x^4+3*x^3+2*x^2+2*x+1) / ((x-1)^3*(x+1)).
%F A253688 (End)
%o A253688 (PARI)
%o A253688 {
%o A253688 a=1;d1=0;p=a;print1(a,", ");\\5v3b
%o A253688 for(n=2,100,
%o A253688    if(n<3,d1=2,
%o A253688      if(n<4,d1=3,
%o A253688        if(n<5,d1=5,
%o A253688          if(n<6,d1=7,
%o A253688            if(n<7,d1=7,
%o A253688              if(n<8,d1=5,
%o A253688                if(Mod(n,2)==0,d1=5,d1=7
%o A253688                )
%o A253688              )
%o A253688            )
%o A253688          )
%o A253688        )
%o A253688      )
%o A253688    );
%o A253688    a=a+d1;p=p+a;
%o A253688    print1(p,", ")
%o A253688 )
%o A253688 }
%Y A253688 Cf. A253687 (side-to-side).
%K A253688 nonn
%O A253688 1,2
%A A253688 _Kival Ngaokrajang_, Jan 09 2015