A261957 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 "vertex to side" 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.

1, 3, 9, 12, 24, 12, 24, 18, 36, 33, 57, 45, 81, 36, 78, 42, 90, 57, 111, 69, 135, 60, 132, 66, 144, 81, 165, 93, 189, 84, 186, 90, 198, 105, 219, 117, 243, 108, 240, 114, 252, 129, 273, 141, 297, 132, 294, 138, 306, 153

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 A261951 A261952 |

| S-V A261950 A008486 A008486 A261956 |

| V-S A249246 A008486 A008486 a(n) |

| 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=24; o=12; print1("1, 3, 9, 12, 24, ", o, ", ", e, ", "); for(n=7, 100, if (Mod(n,2)==0, if (Mod(n,8)==0, e=e+12); if (Mod(n,8)==2, e=e+21); if (Mod(n,8)==4, e=e+24); if (Mod(n,8)==6, e=e-3); print1(e, ", "), if (Mod(n,8)==7, o=o+6); if (Mod(n,8)==1, o=o+15); if (Mod(n,8)==3, o=o+12); if (Mod(n,8)==5, o=o-9); print1(o, ", ")))}

Formula

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

a(n) = a(n-2)+a(n-8)-a(n-10) for n>14.

G.f.: (3*x^14+9*x^13-9*x^12-3*x^11-13*x^10-12*x^9-11*x^8-6*x^7-15*x^4-9*x^3-8*x^2-3*x-1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).

(End)