A261955 - OEIS

%I #19 Sep 16 2015 17:13:31

%S 1,3,6,15,12,24,15,33,21,45,39,72,36,78,39,87,45,99,63,126,60,132,63,

%T 141,69,153,87,180,84,186,87,195,93,207,111,234,108,240,111,249,117,

%U 261,135,288,132,294,135,303,141,315,159

%N Start with a single equilateral triangle for n=0; for the odd n-th generation add a triangle at each expandable side of the triangles of the (n-1)-th generation (this is the "side to side" 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 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  A261956 |

%C |      V-S           A249246  A008486  A008486  A261957 |

%C |      S-S           A261953  A261954    a(n)   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="/A261955/a261955.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>13.

%F G.f.: -(3*x^13+9*x^12-15*x^11-13*x^10-9*x^9-5*x^8-9*x^7-3*x^6-9*x^5-6*x^4-12*x^3-5*x^2-3*x-1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).

%F (End)

%o (PARI) {e=12; o=24; print1("1, 3, 6, 15, ", 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+9); if (Mod(n,8)==1, o=o+12); if (Mod(n,8)==3, o=o+27); if (Mod(n,8)==5, o=o+6); print1(o, ", ")))}

%Y Cf. A008486, A248969, A249246.

%K nonn

%O 0,2

%A _Kival Ngaokrajang_, Sep 06 2015