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.

1, 3, 9, 12, 24, 24, 39, 27, 54, 33, 69, 42, 84, 54, 99, 57, 114, 63, 129, 72, 144, 84, 159, 87, 174, 93, 189, 102, 204, 114, 219, 117, 234, 123, 249, 132, 264, 144, 279, 147, 294, 153, 309, 162, 324, 174, 339, 177, 354

Offset: 0

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

• nonn

Comments

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

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

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

+-------------------------------------------------------+

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

+-------------------------------------------------------+

| Odd n-th version |

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

| S-V A261950 A008486 A008486 A261956 |

| V-S A249246 A008486 A008486 A261957 |

| S-S A261953 A261954 A261955 A008486 |

+-------------------------------------------------------+

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

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

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A008486, A248969, A249246.

Programs

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, ", ")))}

Formula

Conjectures from Colin Barker, Sep 10 2015: (Start)

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

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)).

(End)