A008486 -id:A008486 - cp's OEIS frontend

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.

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

Kival Ngaokrajang, Sep 06 2015

nonn

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.

Kival Ngaokrajang, Illustration of initial terms

Cf. A008486, A248969, A249246.

{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, ", ")))}

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)

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

Kival Ngaokrajang, Sep 06 2015

nonn

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.

Kival Ngaokrajang, Illustration of initial terms

Cf. A008486, A248969, A249246.

{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, ", ")))}

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

Kival Ngaokrajang, Sep 06 2015

nonn

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).

Kival Ngaokrajang, Illustration of initial terms

Cf. A008486, A248969, A249246, A245094.

{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, ", "))}

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

Kival Ngaokrajang, Sep 06 2015

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.

Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A008486, A017197, A248969, A249246.

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, ", ")))

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

Kival Ngaokrajang, Sep 06 2015

nonn

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.

Kival Ngaokrajang, Illustration of initial terms

Cf. A008486, A248969, A249246.

{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, ", ")))}

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

Kival Ngaokrajang, Sep 06 2015

nonn

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.

Kival Ngaokrajang, Illustration of initial terms

Cf. A008486, A248969, A249246.

{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, ", ")))}

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

Kival Ngaokrajang, Sep 06 2015

nonn

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.

Kival Ngaokrajang, Illustration of initial terms

Cf. A008486, A248969, A249246.

{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, ", ")))}

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)

A329508 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n hexagons cut from the hexagonal grid by cuts parallel to grid lines.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 3, 5, 2, 1, 3, 6, 5, 2, 1, 3, 6, 8, 4, 2, 1, 3, 6, 9, 8, 4, 2, 1, 3, 6, 9, 11, 7, 4, 2, 1, 3, 6, 9, 12, 11, 6, 4, 2, 1, 3, 6, 9, 12, 14, 10, 6, 4, 2, 1, 3, 6, 9, 12, 15, 14, 9, 6, 4, 2, 1, 3, 6, 9, 12, 15, 17, 13, 8, 6, 4, 2

Offset: 1

N. J. A. Sloane, Nov 22 2019

This is the structure of carbon nanotubes.
For the case when the cuts are perpendicular to the grid lines, see A329512 and A329515.
See A329501 and A329504 for coordination sequences for cylinders formed by rolling up the square grid.

			Array begins:
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
1, 3, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, ...
1, 3, 6, 8, 8, 7, 6, 6, 6, 6, 6, 6, ...
1, 3, 6, 9, 11, 11, 10, 9, 8, 8, 8, 8, ...
1, 3, 6, 9, 12, 14, 14, 13, 12, 11, 10, 10, ...
1, 3, 6, 9, 12, 15, 17, 17, 16, 15, 14, 13, ...
1, 3, 6, 9, 12, 15, 18, 20, 20, 19, 18, 17, ...
1, 3, 6, 9, 12, 15, 18, 21, 23, 23, 22, 21, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 26, 26, 25, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 29, 29, ...
The initial antidiagonals are:
1
1,2
1,3,2
1,3,5,2
1,3,6,5,2
1,3,6,8,4,2
1,3,6,9,8,4,2
1,3,6,9,11,7,4,2
1,3,6,9,12,11,6,4,2
1,3,6,9,12,14,10,6,4,2
...

Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
N. J. A. Sloane, Illustration for rows 1 through 3, showing vertices of cylinder labeled with distance from base point 0 (c = n is the width (or circumference)). The cylinders are formed by identifying the black lines.
N. J. A. Sloane, Illustration for row 4, showing vertices of cylinder labeled with distance from base point 0 (c = n is the width (or circumference)). The cylinder is formed by identifying the black lines. The trunks are colored blue, the branches red, and the twigs green.
N. J. A. Sloane, Illustration for row 5, showing vertices of cylinder labeled with distance from base point 0 (c = n is the width (or circumference)). The cylinder is formed by identifying the black lines. The trunks are colored blue, the branches red, and the twigs green.
Index entries for coordination sequences

Rows 1,2,3,4 are A040000, A329509, A329510, A329511.

Cf. A008486, A329501-A329517.

c := 4; \\ set c
R := RationalFunctionField(Integers());
FG3 := FreeGroup(3);
Q3 := quo;
H := AutomaticGroup(Q3);
f3 := GrowthFunction(H);
PSR := PowerSeriesRing(Integers():Precision := 60);
Coefficients(PSR!f3);
// 1, 3, 6, 9, 11, 11, 10, 9, 8, 8, 8, 8, 8, 8, 8, ... (row c)
f3;  // g.f. for row c
// (x^8 + x^7 + x^6 - 2*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1)
// = (1+x)*(x^3-x-1)*(x^2+1)^2/(x-1)

The g.f.s for the rows were found and proved using the "trunks and branches" method (see Goodman-Strauss and Sloane). In the illustrations for n=4 and n=5, the trunks are colored blue, the branches red, and the twigs green.
The g.f. G(c) for row c (c>=1) is
(1/(1-x))*(1 + 2*x + 3*x^2*(1-x^(c-2))/(1-x) + 2*x^c - x^(c+2)*(1-x^(c-1))/(1-x)).
The values of G(1) through G(8) are:
(1+x)/(1-x),
(1+x)*(x^3-x^2-x-1)/(x-1),
(1+x)*(x^2+x+1)*(x^3-x^2-1)/(x-1),
(1+x)*(x^3-x-1)*(x^2+1)^2/(x-1),
(1+x)*(x^4+x^3+x^2+x+1)*(x^5-x^4+x^3-x^2-1)/(x-1),
(1+x)*(x^2+x+1)*(x^2-x+1)*(x^7-x^2-x-1)/(x-1),
(1+x)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x^7-x^6+x^5-x^4+x^3-x^2-1)/(x-1),
(1+x)*(x^7-x^5+x^3-x-1)*(x^4+1)*(x^2+1)^2/(x-1).
Note that row n is equal to 2*n once the 2*n-th term has been reached.
The g.f.s for the rows can also be calculated by regarding the 1-skeleton of the cylinder as the Cayley diagram for an appropriate group H, and computing the growth function for H (see the MAGMA code).

A191671 Dispersion of A004772 (>1 and congruent to 0 or 2 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 7, 9, 4, 10, 12, 13, 6, 14, 16, 18, 17, 8, 19, 22, 24, 23, 21, 11, 26, 30, 32, 31, 28, 25, 15, 35, 40, 43, 42, 38, 34, 29, 20, 47, 54, 58, 56, 51, 46, 39, 33, 27, 63, 72, 78, 75, 68, 62, 52, 44, 37, 36, 84, 96, 104, 100, 91, 83, 70, 59, 50, 41

Offset: 1

Clark Kimberling, Jun 11 2011

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
   a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
   a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

			Northwest corner:
1....2....3....4....6
5....7....10...14...19
9....12...16...22...30
13...18...24...32...43
17...23...31...42...56

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A016813, A004772, A191667, A191452.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 3; c2 = 4; m[n_] := If[Mod[n, 3] == 0, 1, 0];
f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
Table[f[n], {n, 1, 30}]  (* A004772 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191671 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191671  *)

A191672 Dispersion of A042965 (>1 and congruent to 0 or 1 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 7, 9, 10, 12, 11, 13, 15, 14, 17, 16, 19, 21, 20, 18, 24, 23, 27, 29, 28, 25, 22, 33, 32, 37, 40, 39, 35, 31, 26, 45, 44, 51, 55, 53, 48, 43, 36, 30, 61, 60, 69, 75, 72, 65, 59, 49, 41, 34, 83, 81, 93, 101, 97, 88, 80, 67, 56, 47, 38

Offset: 1

Clark Kimberling, Jun 11 2011

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
   a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
   a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

			Northwest corner:
1....3...5....8....12
2....4...7....11...16
6....9...13...19...27
10...15..21...29...40
14...20..28...39...53

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A016825, A042965, A191668, A191452.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a = 3; b = 4; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
Table[f[n], {n, 1, 30}] (* A042965 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191672 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191672 *)

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A329508 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n hexagons cut from the hexagonal grid by cuts parallel to grid lines.

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A191671 Dispersion of A004772 (>1 and congruent to 0 or 2 or 3 mod 4), by antidiagonals.

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