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Start with a single equilateral triangle; at odd n-th generation add a similar triangle at each expandable vertex (this is the "vertex to vertex" version); alternate with the "side to vertex" version for even n-th generation; a(n) is the number of triangle for each generation.
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%I #9 Nov 10 2014 17:19:50

%S 1,3,6,15,18,42,24,57,30,72,36,87,48,114,54,129,60,144,66,159,78,186,

%T 84,201,90,216,96,231,108,258,114,273,120,288,126,303,138,330,144,345,

%U 150,360,156,375,168,402,174,417,180,432,186,447,198,474,204,489,210,504,216,519

%N Start with a single equilateral triangle; at odd n-th generation add a similar triangle at each expandable vertex (this is the "vertex to vertex" version); alternate with the "side to vertex" version for even n-th generation; a(n) is the number of triangle for each generation.

%C The construction rules alternate between "vertex to vertex" (A061777 & companions) and "side to vertex" (A101946 & companions). See the link for an illustration.

%H Kival Ngaokrajang, <a href="/A248969/a248969.pdf">Illustration of initial terms</a>

%F Empirical g.f.: (3*x^11 +x^10 +12*x^9 +5*x^8 +15*x^7 +6*x^6 +27*x^5 +12*x^4 +12*x^3 +5*x^2 +3*x +1) / ((x -1)^2*(x +1)^2*(x^2 +1)*(x^4 +1)). - _Colin Barker_, Oct 18 2014

%o (PARI)

%o {

%o c2=0;c3=0;c6=3;c7=1;c8=0;

%o for(n=0,100,

%o if (Mod(n,2)==0,

%o \\even

%o if (n<1,a(n)=1,c3=c3+c2;a=6*c3);

%o c1=n/8+3/4;

%o if (c1==floor(c1),c2=2,c2=1)

%o ,

%o \\odd

%o c4=(n^2-1)/16;

%o if (c4==floor(c4),c5=-1,c5=1);

%o if (n>4, c6=c6+c5);

%o if (n>=2, c7=c7+c6);

%o if (c6<>4, c9=0,c9=2);

%o a=3*(c7+c8+c9);

%o c8=c7

%o );

%o print1(a", ")

%o )

%o }

%Y Vertex to vertex: A061777, A247618, A247619, A247620.

%Y Side to side: A101946, A247903, A247904, A247905.

%K nonn

%O 0,2

%A _Kival Ngaokrajang_, Oct 18 2014

%E Edited. Small changes in the text. - _Wolfdieter Lang_, Nov 10 2014