A261958 - OEIS

%I #26 Oct 12 2015 16:18:20

%S 1,4,12,16,24,32,28,36,32,44,44,56,56,72,60,76,64,84,76,96,88,112,92,

%T 116,96,124,108,136,120,152,124,156,128,164,140,176,152,192,156,196,

%U 160,204,172,216,184,232,188,236,192,244,204,256,216

%N Start with a single square for n=0; for the odd n-th generation add a square at each expandable vertex of the squares of the (n-1)-th generation (this is the "vertex to vertex" version); for the even n-th generation use the "side to vertex" version; a(n) is the number of squares 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           A008574   a(n)      ...      ...   |

%C |      S-V             ...    A008574  A008574    ...   |

%C |      V-S             ...    A008574  A008574    ...   |

%C |      S-S             ...      ...      ...    A008574 |

%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="/A261958/a261958.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.: (x^10+4*x^9+3*x^8+4*x^7+4*x^6+16*x^5+12*x^4+12*x^3+11*x^2+4*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).

%F (End)

%o (PARI) {e=12; o=4; print1("1, ", o, ", ", e, ", "); for(n=3, 100, if (Mod(n,2)==0, if (Mod(n,8)==4, e=e+12); if (Mod(n,8)==6, e=e+4); if (Mod(n,8)==0, e=e+4); if (Mod(n,8)==2, e=e+12); print1(e, ", "), if (Mod(n,8)==3, o=o+12); if (Mod(n,8)==5, o=o+16); if (Mod(n,8)==7, o=o+4); if (Mod(n,8)==1, o=o+8); print1(o, ", ")))}

%Y Cf. A008574, A047557, A249246.

%K nonn

%O 0,2

%A _Kival Ngaokrajang_, Sep 06 2015