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A261951 Start with a single equilateral triangle for n=0; for the odd n-th generation add a triangle at each expandable vertex of the triangles of the (n-1)-th generation (this is the "vertex to vertex" version); for the even n-th generation use the "vertex to side" version; a(n) is the number of triangles added in the n-th generation. 8

%I

%S 1,3,9,12,24,24,39,27,54,33,69,42,84,54,99,57,114,63,129,72,144,84,

%T 159,87,174,93,189,102,204,114,219,117,234,123,249,132,264,144,279,

%U 147,294,153,309,162,324,174,339,177,354

%N Start with a single equilateral triangle for n=0; for the odd n-th generation add a triangle at each expandable vertex of the triangles of the (n-1)-th generation (this is the "vertex to vertex" version); for the even n-th generation use the "vertex to side" version; a(n) is the number of triangles added in the n-th generation.

%C See a comment on V-V and V-S at A249246.

%C The overlap rules for the expansion are: (i) overlap within generation is allowed. (ii) overlap of different generations is prohibited.

%C There are a total of 16 combinations as shown in the table below:

%C +-------------------------------------------------------+

%C | Even n-th version V-V S-V V-S S-S |

%C +-------------------------------------------------------+

%C | Odd n-th version |

%C | V-V A008486 A248969 a(n) A261952 |

%C | S-V A261950 A008486 A008486 A261956 |

%C | V-S A249246 A008486 A008486 A261957 |

%C | S-S A261953 A261954 A261955 A008486 |

%C +-------------------------------------------------------+

%C Note: V-V = vertex to vertex, S-V = side to vertex,

%C V-S = vertex to side, S-S = side to side.

%H Kival Ngaokrajang, <a href="/A261951/a261951.pdf">Illustration of initial terms</a>

%F Conjectures from _Colin Barker_, Sep 10 2015: (Start)

%F a(n) = a(n-2)+a(n-8)-a(n-10) for n>10.

%F G.f.: (7*x^10+3*x^9+14*x^8+3*x^7+15*x^6+12*x^5+15*x^4+9*x^3+8*x^2+3*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).

%F (End)

%o (PARI) {e=9; o=3; print1("1, ", o, ", ", e, ", "); for(n=3, 100, if (Mod(n,2)==0, e=e+15; print1(e, ", "), if (Mod(n,8)==3, o=o+9); if (Mod(n,8)==5, o=o+12); if (Mod(n,8)==7, o=o+3); if (Mod(n,8)==1, o=o+6); print1(o, ", ")))}

%Y Cf. A008486, A248969, A249246.

%K nonn

%O 0,2

%A _Kival Ngaokrajang_, Sep 06 2015

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Last modified June 22 21:29 EDT 2021. Contains 345393 sequences. (Running on oeis4.)