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A261953 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 "side to side" 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. 9
1, 3, 9, 12, 18, 21, 27, 30, 36, 39, 45, 48, 54, 57, 63, 66, 72, 75, 81, 84, 90, 93, 99, 102, 108, 111, 117, 120, 126, 129, 135, 138, 144, 147, 153, 156, 162, 165, 171, 174, 180, 183, 189, 192, 198, 201, 207, 210, 216, 219, 225, 228, 234, 237, 243, 246, 252 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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  A261957 |

|      S-S             a(n)   A261954  A261955  A008486 |

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

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

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

From Manfred Boergens, Sep 21 2021: (Start)

For finite sets of random points in the real plane with exactly n nearest neighbors, a(n) for n >= 2 is a lower bound for the maximal number of points. Conjecturally, a(n) equals this number.

The randomness provides for pairwise different distances with probability = 1.

A point A is called a nearest neighbor if there is a point B with smaller distance to A than to any other point C.

In graph theory terms: Let G be a finite simple digraph; the vertices of G are arbitrary placed points in R^2 with pairwise different distances; the edges of G are arrows joining each point (tail end) to its nearest neighbor (head end). If G contains exactly n nearest neighbors and b(n) is the maximal number of points in any such graph then a(n) is the best lower bound known as yet for b(n).

a(n) for n >= 2 can be seen as an "inverse" to A347941.

a(n) is built by constructing G with m points and n nearest neighbors, m chosen as maximal as possible, then defining a(n)=m. The start is a(2)=9 and a(3)=12. We call the pairs (m,n)=(9,2) and (m,n)=(12,3) "anchor pairs" and proceed to bigger n by combining graphs with these anchor pairs to bigger graphs. So the next anchor pairs are (18,4), (21,5) and (27,6).

We conjecture that a(n) is optimal. This claim is true if the following assumptions hold:

- The anchor pairs (9,2) and (12,3) are optimal.

- All bigger anchor pairs (m,n) are constructed by combining copies of (9,2) if n is even and adding one (12,3) if n is odd.

(End)

LINKS

Table of n, a(n) for n=0..56.

Kival Ngaokrajang, Illustration of initial terms

Manfred Boergens, Next-neighbours

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

FORMULA

a(0)=1, a(1)=3; for n >= 2, a(n) = a(n-1) + 6, if mod(n,2) = 0, otherwise a(n) = a(n-1) + 3.

From Colin Barker, Sep 10 2015: (Start)

a(n) = (3*(-1+(-1)^n+6*n))/4.

a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.

G.f.: (x^3+5*x^2+2*x+1) / ((x-1)^2*(x+1)).

(End)

a(n) = 3 * A032766(n) for n>=1. - Michel Marcus, Sep 13 2015

a(0)=1; for n >= 1, a(n) = 9n/2 for even n, a(n) = (9n-3)/2 for odd n. - Manfred Boergens, Sep 21 2021

EXAMPLE

If the graph G in the comment by Manfred Boergens has 5 nearest neighbors there are at most 21 vertices in G (conjectured; it is proved that there are G with 5 nearest neighbors and 21 vertices but it is not yet proved that 21 is the maximum). - Manfred Boergens, Sep 21 2021

MATHEMATICA

Join[{1}, Table[If[OddQ[n], (9 n - 3)/2, 9 n/2], {n, 1, 100}]] - Manfred Boergens, Sep 21 2021

PROG

(PARI) {a=3; print1("1, ", a, ", "); for(n=2, 100, if (Mod(n, 2)==0, a=a+6, a=a+3); print1(a, ", "))}

CROSSREFS

Cf. A032766, A008486, A248969, A249246, A347941.

Sequence in context: A274676 A310321 A310322 * A297001 A309394 A285564

Adjacent sequences:  A261950 A261951 A261952 * A261954 A261955 A261956

KEYWORD

nonn,easy

AUTHOR

Kival Ngaokrajang, Sep 06 2015

STATUS

approved

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Last modified July 5 23:40 EDT 2022. Contains 355108 sequences. (Running on oeis4.)