A261952 - OEIS

%I #18 Sep 14 2015 17:10:32

%S 1,3,9,18,18,24,27,33,36,42,45,51,54,60,63,69,72,78,81,87,90,96,99,

%T 105,108,114,117,123,126,132,135,141,144,150,153,159,162,168,171,177,

%U 180,186,189,195,198,204,207,213,216,222

%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 "side 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 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  A261951    a(n)  |

%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.

%C For n > 4, a(n) = A245094(n+1).

%H Kival Ngaokrajang, <a href="/A261952/a261952.pdf">Illustration of initial terms</a>

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

%F a(n) = 3*(1-(-1)^n+6*n)/4 for n>3.

%F a(n) = a(n-1)+a(n-2)-a(n-3) for n>6.

%F G.f.: (3*x^6-3*x^5-6*x^4+7*x^3+5*x^2+2*x+1) / ((x-1)^2*(x+1)).

%F (End)

%o (PARI) {a=18; print1("1, 3, 9, 18, ", a, ", "); for(n=5, 100, if (Mod(n,2)==0, a=a+3, a=a+6); print1(a, ", "))}

%Y Cf. A008486, A248969, A249246, A245094.

%K nonn

%O 0,2

%A _Kival Ngaokrajang_, Sep 06 2015