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 1-7 of 7 results.

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.

Original entry on oeis.org

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
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 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)

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    Join[{1}, Table[If[OddQ[n], (9 n - 3)/2, 9 n/2], {n, 1, 100}]] (* Manfred Boergens, Sep 21 2021 *)
  • PARI
    {a=3; print1("1, ", a, ", "); for(n=2, 100, if (Mod(n,2)==0, a=a+6, a=a+3); print1(a, ", "))}

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

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)
Showing 1-7 of 7 results.

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