cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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.

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

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

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

A261951 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

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 a(n) A261952 |
| S-V A261950 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

Crossrefs

Cf. A008486, A248969, A249246.

Programs

  • 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, ", ")))}

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.: (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)

A261952 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

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 a(n) |
| S-V A261950 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.
For n > 4, a(n) = A245094(n+1).

Links

Crossrefs

Cf. A008486, A248969, A249246, A245094.

Programs

  • 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, ", "))}

Formula

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

A261954 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

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 A261953 a(n) A261955 A008486 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.

Links

Crossrefs

Cf. A008486, A017197, A248969, A249246.

Programs

  • 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, ", ")))

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.
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+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)

A261955 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

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 A261953 A261954 a(n) A008486 |
+-------------------------------------------------------+
Note: V-V = vertex to vertex, S-V = side to vertex,
V-S = vertex to side, S-S = side to side.

Links

Crossrefs

Cf. A008486, A248969, A249246.

Programs

  • 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, ", ")))}

Formula

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

A261956 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

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 a(n) |
| 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

Crossrefs

Cf. A008486, A248969, A249246.

Programs

  • 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, ", ")))}

Formula

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

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.

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, Sep 06 2015

Keywords

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

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)

A347941 For sets of n random points in the real plane, a(n) is an upper bound for the minimal number of nearest neighbors.

Original entry on oeis.org

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
Offset: 2

Views

Author

Manfred Boergens, Sep 20 2021

Keywords

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.

Examples

			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).

Links

Crossrefs

Cf. A261953.

Programs

  • Mathematica
    h=(n+5)/9; Join[{2,2}, Table[2 Floor[h] + If[FractionalPart[h]<2/3, 0, 1], {n, 4, 100}]]

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
Showing 1-8 of 8 results.

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