A261958 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.

1, 4, 12, 16, 24, 32, 28, 36, 32, 44, 44, 56, 56, 72, 60, 76, 64, 84, 76, 96, 88, 112, 92, 116, 96, 124, 108, 136, 120, 152, 124, 156, 128, 164, 140, 176, 152, 192, 156, 196, 160, 204, 172, 216, 184, 232, 188, 236, 192, 244, 204, 256, 216, 272, 220, 276, 224, 284

Offset

0,2

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 A008574 a(n) ... ... |

| S-V ... A008574 A008574 ... |

| V-S ... A008574 A008574 ... |

| S-S ... ... ... A008574 |

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

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

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

Links

Table of n, a(n) for n=0..57.

Jorge Calero-Sanz and Luis Felipe Prieto-Martínez, Roots in the substitution group and in the group of Riordan matrices with ones in the main diagonal, Linear Algebra and its Applications 747, 234-253 (2026), see p. 247; arXiv preprint, arXiv:2509.25972 [math.GR], 2025. See p. 16.

Kival Ngaokrajang, Illustration of initial terms

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.: (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)).

(End)

Prog

(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, ", ")))}

Crossrefs

Cf. A008574, A047557, A249246.

Sequence in context: A393044 A053006 A328849 * A057962 A186303 A073687

Adjacent sequences: A261955 A261956 A261957 * A261959 A261960 A261961

Keyword

nonn,changed

Author

Kival Ngaokrajang, Sep 06 2015

Extensions

a(53)-a(57) from Stefano Spezia, Oct 04 2025

Status

approved