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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.
+10
8
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
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 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.
FORMULA
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)
PROG
(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, ", ")))}
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.
+10
8
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
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 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.
FORMULA
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)
PROG
(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, ", ")))}
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 "side to side" version; a(n) is the number of triangles added in the n-th generation.
+10
8
1, 3, 9, 18, 18, 24, 27, 33, 36, 42, 45, 51, 54, 60, 63, 69, 72, 78, 81, 87, 90, 96, 99, 105, 108, 114, 117, 123, 126, 132, 135, 141, 144, 150, 153, 159, 162, 168, 171, 177, 180, 186, 189, 195, 198, 204, 207, 213, 216, 222
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 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.
FORMULA
a(n) = 3*(1-(-1)^n+6*n)/4 for n>3.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>6.
G.f.: (3*x^6-3*x^5-6*x^4+7*x^3+5*x^2+2*x+1) / ((x-1)^2*(x+1)).
(End)
PROG
(PARI) {a=18; print1("1, 3, 9, 18, ", a, ", "); for(n=5, 100, if (Mod(n, 2)==0, a=a+3, a=a+6); print1(a, ", "))}
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 "side to vertex" version; a(n) is the number of triangles added in the n-th generation.
+10
8
1, 3, 3, 6, 12, 15, 21, 18, 30, 27, 39, 30, 48, 39, 57, 42, 66, 51, 75, 54, 84, 63, 93, 66, 102, 75, 111, 78, 120, 87, 129, 90, 138, 99, 147, 102, 156, 111, 165, 114, 174, 123, 183, 126, 192, 135, 201, 138, 210, 147, 219
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 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.
FORMULA
a(0) = 1, a(1) = 3; for even n >= 2, a(n) = 9*(n/2-1) + 3 or a(n) = A017197(n/2-1); for odd n >= 3, a(n) = a(n-2) + 9, if mod(n,4) = 1 otherwise a(n) = a(n-2) + 3. a(n) = a(n-2)+a(n-4)-a(n-6) for n>6.
G.f.: (7*x^6+6*x^5+8*x^4+3*x^3+2*x^2+3*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)).
(End)
PROG
(PARI) a=3; print1("1, ", a, ", "); for (n=2, 100, if (Mod(n, 4)==0||Mod(n, 4)==2, print1(9*(n/2-1)+3, ", "), if (Mod(n, 4)==1, a=a+9, a=a+3); print1(a, ", ")))
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 side" version; a(n) is the number of triangles added in the n-th generation.
+10
8
1, 3, 6, 15, 12, 24, 15, 33, 21, 45, 39, 72, 36, 78, 39, 87, 45, 99, 63, 126, 60, 132, 63, 141, 69, 153, 87, 180, 84, 186, 87, 195, 93, 207, 111, 234, 108, 240, 111, 249, 117, 261, 135, 288, 132, 294, 135, 303, 141, 315, 159
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 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.
FORMULA
a(n) = a(n-2)+a(n-8)-a(n-10) for n>13.
G.f.: -(3*x^13+9*x^12-15*x^11-13*x^10-9*x^9-5*x^8-9*x^7-3*x^6-9*x^5-6*x^4-12*x^3-5*x^2-3*x-1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).
(End)
PROG
(PARI) {e=12; o=24; print1("1, 3, 6, 15, ", e, ", ", o, ", "); for(n=6, 100, if (Mod(n, 2)==0, if (Mod(n, 8)==6, e=e+3); if (Mod(n, 8)==0, e=e+6); if (Mod(n, 8)==2, e=e+18); if (Mod(n, 8)==4, e=e-3); Print1(e, ", "), if (Mod(n, 8)==7, o=o+9); if (Mod(n, 8)==1, o=o+12); if (Mod(n, 8)==3, o=o+27); if (Mod(n, 8)==5, o=o+6); print1(o, ", ")))}
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 "side to side" version; a(n) is the number of triangles added in the n-th generation.
+10
8
1, 3, 6, 9, 12, 18, 15, 21, 21, 36, 39, 54, 36, 54, 39, 57, 45, 72, 63, 90, 60, 90, 63, 93, 69, 108, 87, 126, 84, 126, 87, 129, 93, 144, 111, 162, 108, 162, 111, 165, 117, 180, 135, 198, 132, 198, 135, 201, 141, 216, 159
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 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.
FORMULA
G.f.: -(9*x^13 +9*x^12 -12*x^11 -13*x^10 -12*x^9 -5*x^8 -3*x^7 -3*x^6 -9*x^5 -6*x^4 -6*x^3 -5*x^2 -3*x -1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).
a(n) = a(n-2) + a(n-8) - a(n-10) for n > 13. (End)
PROG
(PARI) {e=12; o=18; print1("1, 3, 6, 9, ", e, ", ", o, ", "); for(n=6, 100, if (Mod(n, 2)==0, if (Mod(n, 8)==6, e=e+3); if (Mod(n, 8)==0, e=e+6); if (Mod(n, 8)==2, e=e+18); if (Mod(n, 8)==4, e=e-3); print1(e, ", "), if (Mod(n, 8)==7, o=o+3); if (Mod(n, 8)==1, o=o+15); if (Mod(n, 8)==3, o=o+18); if (Mod(n, 8)==5, o=o+0); print1(o, ", ")))}
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.
+10
8
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
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 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.
FORMULA
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)
PROG
(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, ", ")))}
For sets of n random points in the real plane, a(n) is an upper bound for the minimal number of nearest neighbors.
+10
1
2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22, 22, 22, 23
COMMENTS
The sequence deals with sets of n points with pairwise different distances. The randomness in the definition 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 simple digraph; the vertices of G are n arbitrarily 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). Let b(n) be the minimal number of points receiving arrowheads in any such graph. a(n) is the best upper bound yet known for b(n).
A261953(n) for n >= 2 can be seen as an "inverse" to a(n).
a(n) is built by constructing G with n points and m nearest neighbors, m chosen as minimal as possible, then defining a(n)=m.
The start is a(n)=2 for n <= 9 and a(n)=3 for n=10,11,12. We call the pairs (n,m)=(9,2) and (n,m)=(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).
If (n0,m-1) and (n1,m) are anchor pairs then a(n')=m for n0 < n' <= n1.
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 (n,m) are constructed by combining copies of (9,2) if m is even and adding one (12,3) if m is odd.
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).
FORMULA
a(2) = a(3) = 2.
a(n) = 2j for n = 9j-5 ... 9j, j > 0;
a(n) = 2j+1 for n = 9j+1 ... 9j+3, j > 0;
With h=(n+5)/9 for n>3:
a(n) = 2*floor(h) if h-floor(h)<2/3;
a(n) = 2*floor(h)+1 otherwise.
G.f.: -x^2*(x^11-2*x^9+x^8+2)/(-x^10+x^9+x-1). - Alois P. Heinz, Sep 20 2021
EXAMPLE
G with 25 vertices has at least 6 nearest neighbors (conjectured; it is proved that there are G with n=25 and m=6 but it is not yet proved that 6 is the minimum).
MATHEMATICA
h=(n+5)/9; Join[{2, 2}, Table[2 Floor[h] + If[FractionalPart[h]<2/3, 0, 1], {n, 4, 100}]]
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