A261950 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 "vertex to vertex" version; a(n) is the number of triangles added in the n-th generation.

1, 3, 9, 12, 30, 18, 45, 27, 66, 33, 81, 42, 102, 48, 117, 57, 138, 63, 153, 72, 174, 78, 189, 87, 210, 93, 225, 102, 246, 108, 261, 117, 282, 123, 297, 132, 318, 138, 333, 147, 354, 153, 369, 162, 390, 168, 405

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 a(n) 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, if (Mod(n,4)==0, e=e+21); if (Mod(n,4)==2, e=e+15); print1(e, ", "), if (Mod(n,4)==3, o=o+9); if (Mod(n,4)==1, o=o+6); print1(o, ", ")))}

Formula

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

a(n) = a(n-2)+a(n-4)-a(n-6) for n>6.

G.f.: (7*x^6+3*x^5+20*x^4+9*x^3+8*x^2+3*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)).

(End)

Extensions

Typo in data fixed by Colin Barker, Sep 10 2015