A261956 - OEIS

%I #29 Oct 12 2015 16:20:30

%S 1,3,6,9,12,18,15,21,21,36,39,54,36,54,39,57,45,72,63,90,60,90,63,93,

%T 69,108,87,126,84,126,87,129,93,144,111,162,108,162,111,165,117,180,

%U 135,198,132,198,135,201,141,216,159

%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 "side 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 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  A261951  A261952 |

%C |      S-V           A261950  A008486  A008486    a(n)  |

%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="/A261956/a261956.pdf">Illustration of initial terms</a>

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

%F G.f.: -(9*x^13 +9*x^12 -12*x^11 -13*x^10 -12*x^9 -5*x^8 -3*x^7 -3*x^6 -9*x^5 -6*x^4 -6*x^3 -5*x^2 -3*x -1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).

%F a(n) = a(n-2) + a(n-8) - a(n-10) for n > 13. (End)

%o (PARI) {e=12; o=18; print1("1, 3, 6, 9, ", e, ", ", o, ", "); for(n=6, 100, if (Mod(n,2)==0, if (Mod(n,8)==6, e=e+3); if (Mod(n,8)==0, e=e+6); if (Mod(n,8)==2, e=e+18); if (Mod(n,8)==4, e=e-3); print1(e, ", "), if (Mod(n,8)==7, o=o+3); if (Mod(n,8)==1,o=o+15); if (Mod(n,8)==3, o=o+18); if (Mod(n,8)==5, o=o+0); print1(o, ", ")))}

%Y Cf. A008486, A248969, A249246.

%K nonn

%O 0,2

%A _Kival Ngaokrajang_, Sep 06 2015