A008458 -id:A008458 - cp's OEIS frontend

A276158 Triangle read by rows: T(n,k) = 6k(n + 1 - k) for 0 < k <= n; for k = 0, T(n,0) = n + 1.

1, 2, 6, 3, 12, 12, 4, 18, 24, 18, 5, 24, 36, 36, 24, 6, 30, 48, 54, 48, 30, 7, 36, 60, 72, 72, 60, 36, 8, 42, 72, 90, 96, 90, 72, 42, 9, 48, 84, 108, 120, 120, 108, 84, 48, 10, 54, 96, 126, 144, 150, 144, 126, 96, 54

Offset: 0

Stefano Maruelli, Aug 22 2016

The row sums of the triangle provide the positive terms of A000578.
Similar triangles can be generated by the formula P(n,k,m) = (Q(k+1,m)-Q(k,m))*(n+1-k), where Q(i,r) = i^r-(i-1)^r, 0 < k <= n, and P(n,0,m) = n+1. T(n,k) is the case m=3, that is T(n,k) = P(n,k,3).
T(9,k) for 0 <= k <= 9 provides the indegrees of the 10 non-leaf nodes of the network graph of the Kaprekar Process on 3 digits when the nodes are listed in numerical order.  Namely, nodes 000, 099, 198, 297, 396, 495, 594, 693, 792, and 891 have indegrees 10, 54, 96, 126, 144, 150, 144, 126, 96, 54, respectively. Result derived empirically. See "Kaprekar Network Graph for 3 Digits". - Norman Whitehead, May 16 2022

			Triangle starts:
----------------------------------------------
n \ k |  0   1    2    3    4    5    6    7
----------------------------------------------
0     |  1;
1     |  2,  6;
2     |  3, 12,  12;
3     |  4, 18,  24,  18;
4     |  5, 24,  36,  36,  24;
5     |  6, 30,  48,  54,  48,  30;
6     |  7, 36,  60,  72,  72,  60,  36;
7     |  8, 42,  72,  90,  96,  90,  72,  42;
...

Norman Whitehead, Kaprekar Network Graph for 3 Digits
Norman Whitehead, Kaprekar Network Graph Node Count Verification

Cf. A000578, A008458, A276189.

[IsZero(k) select n+1 else 6*k*(n+1-k): k in [0..n], n in [0..10]]; // Bruno Berselli, Aug 31 2016

/* As triangle (see the second comment): */ m:=3; Q:=func; P:=func; [[P(n, k, m): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Aug 31 2016

T:= (n, k) -> `if`(k=0, n+1, 6*k*(n+1-k)):
seq(seq(T(n, k), k=0..n), n=0..30); # Robert Israel, Aug 31 2016

Table[If[k == 0, n + 1, 6 k (n + 1 - k)], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 25 2016 *)

T(n, k) = if (k==0, n+1, 6*k*(n+1-k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 25 2016

Sum_{k=0..n} T(n,k) = T(n,0)^3 = A000578(n+1).
G.f. as triangle: (1+4*x*y + x^2*y^2)/((1-x)^2*(1-x*y)^2). - Robert Israel, Aug 31 2016
T(n,n-h) = (h+1)*A008458(n-h) for 0 <= h <= n. Therefore, the main diagonal of the triangle is A008458. - Bruno Berselli, Aug 31 2016

Corrected and rewritten by Bruno Berselli, Sep 01 2016

A322041 Triangle read by rows: let E denote the standard triangular 6-valent grid in the plane, regarded as a graph with the Eisenstein integers as vertices; row n gives the coordination sequence of the quotient graph E/nE.

Original entry on oeis.org

1, 1, 3, 1, 6, 2, 1, 6, 9, 0, 1, 6, 12, 6, 0, 1, 6, 12, 15, 2, 0, 1, 6, 12, 18, 12, 0, 0, 1, 6, 12, 18, 21, 6, 0, 0, 1, 6, 12, 18, 24, 18, 2, 0, 0, 1, 6, 12, 18, 24, 27, 12, 0, 0, 0, 1, 6, 12, 18, 24, 30, 24, 6, 0, 0, 0, 1, 6, 12, 18, 24, 30, 33, 18, 2, 0, 0, 0, 1, 6, 12, 18, 24, 30, 36, 30, 12, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Dec 05 2018; corrected and extended Dec 06 2018

The Eisenstein integers E are the complex numbers r+s*omega, where r, s in Z and omega = exp(2*Pi*i/3) is a complex cube root of unity.
Denote the entries in the triangle by T(n,k), for n >= 1, 0 <= k <= n-1. Then T(n,k) <= 6*k for k >= 1, and Sum_{k=0..n-1} T(n,k) = n^2.
When E is regarded as a lattice in R^2, E/nE has packing radius roughly n/2, but covering radius roughly n/sqrt(3) > n/2 (see Conway-Sloane, Chapter 4). This means that as n increases, the number of terms in the n-th row of the triangle will increase linearly with n. The largest k such that T(n,k) is nonzero is A322042(n), which is conjecturally n - ceiling(n/3).

			The first 18 rows are
1 [1]
2 [1, 3]
3 [1, 6, 2]
4 [1, 6, 9, 0]
5 [1, 6, 12, 6, 0]
6 [1, 6, 12, 15, 2, 0]
7 [1, 6, 12, 18, 12, 0, 0]
8 [1, 6, 12, 18, 21, 6, 0, 0]
9 [1, 6, 12, 18, 24, 18, 2, 0, 0]
10 [1, 6, 12, 18, 24, 27, 12, 0, 0, 0]
11 [1, 6, 12, 18, 24, 30, 24, 6, 0, 0, 0]
12 [1, 6, 12, 18, 24, 30, 33, 18, 2, 0, 0, 0]
13 [1, 6, 12, 18, 24, 30, 36, 30, 12, 0, 0, 0, 0]
14 [1, 6, 12, 18, 24, 30, 36, 39, 24, 6, 0, 0, 0, 0]
15 [1, 6, 12, 18, 24, 30, 36, 42, 36, 18, 2, 0, 0, 0, 0]
16 [1, 6, 12, 18, 24, 30, 36, 42, 45, 30, 12, 0, 0, 0, 0, 0]
17 [1, 6, 12, 18, 24, 30, 36, 42, 48, 42, 24, 6, 0, 0, 0, 0, 0]
18 [1, 6, 12, 18, 24, 30, 36, 42, 48, 51, 36, 18, 2, 0, 0, 0, 0, 0]
...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. Fig. 7.1, p. 199. Illustrates row 2 (note that E/2E is isomorphic to GF(4)).

N. J. A. Sloane, Rows 1 through 36.
N. J. A. Sloane, Illustration for n=2
N. J. A. Sloane, Illustration for n=3
N. J. A. Sloane, Illustration for n=4
N. J. A. Sloane, Illustration for n=6
N. J. A. Sloane, Illustration for n=8

The rows converge to A008458.

Cf. A322038 (an analog for the square grid), A322042.

# We work in a fundamental region for E/nE and calculate the edge-distance of each point to the nearest point of nE.
hist:=proc(n) local A,i,j,m,d1,d2,d3,d4;
A:=Array(0..n,0);
for i from 0 to n-1 do
for j from 0 to n-1 do
d1:=i+j; d2:=n-i; d3:=2*n-i-j; d4:=n-j;
if i+j

Examination of the first 36 rows (see link) shows an obvious quasi-periodic structure. Call an entry T(n,k) "full" if k=0 or T(n,k)=6*k. Then it appears that column k>0 is full starting at n=2k+1. It also appears that the number of trailing 0's is floor((n-1)/3) (see A322042). Combining these two observations suggests that the rows of the triangle are quasi-periodic with period 6.
One can now formulate a specific conjecture for what row n is, for each of the six residue classes of n mod 6.
For example, suppose n=6t. Then it appears that row n is [1, 6, 18, 24, ..., 18t-6, 18t-3, 18(t-1), 18(t-2), 18(t-3), ..., 36, 18, 2, 0 (2t-1 times)].
For t=3, for example, we get:
[1, 6, 12, 18, 24, 30, 36, 42, 48, 51, 36, 18, 2, 0, 0, 0, 0, 0]
There are similar conjectures for n = 6t+1, ..., 6t+5.

A354958 Coordination sequence for the Manhattan hexagonal lattice with respect to the point (X=0, Y=1).

Original entry on oeis.org

1, 3, 9, 15, 27, 21, 45, 27, 63, 33, 81, 39, 99, 45, 117, 51, 135, 57, 153, 63, 171, 69, 189, 75, 207, 81, 225, 87, 243, 93, 261, 99, 279, 105, 297, 111, 315, 117, 333, 123, 351, 129, 369, 135, 387, 141, 405, 147, 423, 153, 441, 159, 459, 165, 477, 171, 495

Offset: 0

Rémy Sigrist, Jun 13 2022

nonn

In the Manhattan hexagonal lattice, streets parallel to one of the 3 axis have alternating orientations:
               Z         Y
                .---<---.
               / \     / \
              v   ^   ^   v
             /     \ /     \
            .--->---O--->---. X
             \     / \     /
              v   ^   ^   v
               \ /     \ /
                .---<---.

Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, PARI program
Index entries for coordination sequences
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A008458, A336627, A354957.

PARI
```
See Links section.
```

A242941 a(n) is the number of convex uniform tessellations in dimension n.

Original entry on oeis.org

1, 11, 28, 143

Offset: 1

Felix Fröhlich, May 27 2014

Terms for n > 4 have not been determined so far. Alfredo Andreini in 1905 gave a value of 25 for a(3), later found to be incorrect. The value 28 for a(3) was given by Norman Johnson in 1991 and later in 1994 independently by Branko Grünbaum. The value for a(4) was given by George Olshevsky in 2006.
Deza and Shtogrin (2000) agree that the value of a(3) is 28, although the authors do not provide a proof. - Felix Fröhlich, Nov 29 2014
From Felix Fröhlich, Feb 03 2019: (Start)
The 11 convex uniform tilings are all illustrated in Kepler, 1619. For an argument that exactly 11 such tilings exist, see Grünbaum, Shephard, 1977.
In dimension 2, the definition of "uniform polytope" usually seems to be equivalent to the regular polygons in order to exclude polygons that alternate two different edge-lengths. Applying this principle retroactively to dimension 1 (as done, as I assume, by Coxeter, see Coxeter, 1973, p. 129) yields a(1) = 1. (End)

H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486614809.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, Vol. 4, No. 2 (1994), 49-56.
N. W. Johnson, Uniform Polytopes, [To appear, cf. Weiss, Stehle, 2017].

A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle correspondenti reto correlative [On the regular and semiregular nets of polyhedra and on the corresponding correlative nets], Mem. Società Italiana della Scienze, Ser.3, 14 (1905), 75-129.
M. Deza and M. Shtogrin, Uniform Partitions of 3-space, their Relatives and Embedding, European Journal of Combinatorics, Vol. 21, No. 6 (2000), 807-814.
B. Grünbaum and G. C. Shephard, Tilings by Regular Polygons, Mathematics Magazine, Vol. 50, No. 5 (1977), 227-247.
J. Kepler, Harmonices Mundi [The Harmony of the World] (1619).
G. Olshevsky, Uniform Panoploid Tetracombs (2006)
A. I. Weiss and E. M. Stehle, Norman W. Johnson (12 November 1930 to 13 July 2017), The Art of Discrete and Applied Mathematics, Vol. 1, No. 1 (2018).
Wikipedia, Convex uniform honeycomb
Wikipedia, List of convex uniform tilings

Cf. A068599.
List of coordination sequences for the 11 uniform 2D tilings: A008458(the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706(3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for the 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

Edited by N. J. A. Sloane, Feb 15 2018

Edited by Felix Fröhlich, Feb 03-10 2019

A327893 Minesweeper sequence of positive integers arranged in a hexagonal spiral.

Original entry on oeis.org

4, -1, -1, 3, -1, 3, -1, 3, 2, 4, -1, 3, -1, 3, 2, 2, -1, 3, -1, 2, 0, 2, -1, 3, 1, 2, 2, 2, -1, 2, -1, 1, 1, 1, 2, 3, -1, 2, 0, 1, -1, 4, -1, 1, 1, 2, -1, 1, 0, 1, 2, 3, -1, 3, 1, 0, 0, 1, -1, 4, -1, 1, 0, 0, 1, 3, -1, 2, 2, 2, -1, 2, -1, 3, 1, 1, 0, 1, -1, 3

Offset: 1

Michael De Vlieger, Oct 09 2019

Place positive integers on a 2D grid starting with 1 in the center and continue along a hexagonal spiral. Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around it. n is replaced by a(n). This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.

The largest term in the sequence is 4 since 1 is surrounded by 3 odd numbers {3, 5, 7} and the only even prime. Additionally, the pattern of odd and even numbers appears in alternating rows oriented in a triangular symmetry such that no other number has more than four odd numbers. (This courtesy of Witold Tatkiewicz.)

			Consider a spiral grid drawn counterclockwise with the largest number k = A003219(n) = 3*n*(n+1)+1 in "shell" n, and each shell has A008458(n) elements:
          28--27--26--25
          /             \
        29  13--12--11  24
        /   /         \   \
      30  14   4---3  10  23
      /   /   /     \   \   \
    31  15   5   1---2   9  22
      \   \   \         /   /
      32  16   6---7---8  21
        \   \             /
        33  17--18--19--20  ...
          \                /
          34--35--36--37--38
1 is not prime and in the 6 adjacent cells 2 through 7 inclusive, we have 4 primes, therefore a(1) = 4.
2 is prime therefore a(2) = -1.
4 is not prime and in the 6 adjacent cells {1, 3, 12, 13, 14, 5} there are 4 primes, therefore a(4) = 4, etc.
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
         2---2---2---1
        /             \
       *   *---3---*   3
      /   /         \   \
     2   3   3---*   4   *
    /   /   /     \   \   \
   *   2   *   4---*   2   2
    \   \   \         /   /
     1   3   3---*---3   .
      \   \             /
       1   *---3---*---2  ...
        \                 /
         1---2---3---*---2

Michael De Vlieger, Table of n, a(n) for n = 1..10621 (60 concentric hexagons).
Michael De Vlieger, Minesweeper style hexagonal plot of 1261 terms, replacing -1 with n in a black circle, and 0 represented by blank space.
Michael De Vlieger, Hexagonal plot of 30301 terms, (100 concentric hexagons), color coded.
Michael De Vlieger, Hexagonal plot of 120601 terms, (200 concentric hexagons), color coded.
Michael De Vlieger, Plot of 469 terms, with 12 concentric hexagons smoothed into concentric rings, color coded.
Michael De Vlieger, Plot of 120601 terms, with 200 concentric hexagons smoothed into concentric rings, color coded.

Cf. A003215, A008458, A326405, A326406, A326407, A326408, A326409, A326410.

Block[{n = 6, m, s, t, u}, m = n + 1; s = Array[3 #1 (#1 - 1) + 1 + #2 #1 + #3 & @@ {#3, #4, Which[Mod[#4, 3] == 0, Abs[#1], Mod[#4, 3] == 1, Abs[#2], True, Abs[#2] - Abs[#1]]} & @@ {#1, #2, If[UnsameQ @@ Sign[{#1, #2}], Abs[#1] + Abs[#2], Max[Abs[{#1, #2}]]], Which[And[#1 > 0, #2 <= 0], 0, And[#1 >= #2, #1 + #2 > 0], 1, And[#2 > #1, #1 >= 0], 2, And[#1 < 0, #2 >= 0], 3, And[#1 <= #2, #1 + #2 < 0], 4, And[#1 > #2, #1 + #2 <= 0], 5, True, 0]} & @@ {#2 - m - 1, m - #1 + 1} &, {#, #}] &[2 m + 1]; t = s /. k_ /; k > 3 n (n + 1) + 1 :> -k; Table[If[PrimeQ@ m, -1, Count[#, _?PrimeQ] &@ Union@ Map[t[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[t, m] + {##} - 2 & @@ {#1, #2 + Boole[#1 == #2 == 2] + Boole[#1 == 1]} &, {3, 2}]]], {m, 3 n (n - 1) + 1}]]

A086461 Symmetric version of square array A086460.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 4, 6, 6, 4, 1, 1, 5, 8, 9, 8, 5, 1, 1, 6, 10, 12, 12, 10, 6, 1, 1, 7, 12, 15, 16, 15, 12, 7, 1, 1, 8, 14, 18, 20, 20, 18, 14, 8, 1, 1, 9, 16, 21, 24, 25, 24, 21, 16, 9, 1, 1, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 1, 1, 11, 20, 27, 32, 35, 36

Offset: 0

Paul Barry, Jul 21 2003

Rows include A028310, A004277, A008486, A008574, A008706, A008458. Main diagonal is n^2+0^n (A000290, preceded by extra 1).

			Rows begin
  1 1 1 1 1 ...
  1 1 2 3 4 ...
  1 2 4 6 8 ...
  1 3 6 9 12 ...
  1 4 8 12 16 ...
As a triangle:
  {1},
  {1, 1},
  {1, 1, 1},
  {1, 2, 2, 1},
  {1, 3, 4, 3, 1},
  {1, 4, 6, 6, 4, 1},
  {1, 5, 8, 9, 8, 5, 1},
  {1, 6, 10, 12, 12, 10, 6, 1},
  {1, 7, 12, 15, 16, 15, 12, 7, 1},
  {1, 8, 14, 18, 20, 20, 18, 14, 8, 1},
  {1, 9, 16, 21, 24, 25, 24, 21, 16, 9, 1}

t[n_, m_] = If[ n == 0 || n == m || m == 0, 1, n - m]*If[n == m || n == 0 || m == 0, 1, m]; Table[t[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Roger L. Bagula, Sep 06 2008 *)

T(0, k)=T(n, 0)=1, T(n, k)=nk+0^n, n, k>0

Alternatively, triangle read by rows with formula t(n,m)=If[n == 0 || n == m || m == 0, 1, n - m]*If[n == m || n == 0 || m == 0, 1, m]. - Roger L. Bagula, Sep 06 2008

A130702 Possible sides in the Euler V=E-F+2 as roots in a cubic polynomial of the form: P(x)=(x-V)(x-F)(x+E) =x^3+(E-V-F)x^2+(VF-E(V+F))x=EF*V Solves here for F ( Face, Edge, Vertex).

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 64, 66, 72

Offset: 1

Roger L. Bagula, Jul 06 2007

Polynomial cubic of Euler's V,F,E: V=E-F+2 P(x)=(x-V)*(x-F)*(x+E) =x^3+(E-V-F)*x^2+(V*F-E(V+F))*x=E*F*V letting (E-V-F)=-2 V+F=E+2 and product: p=V*F I got P(x)=x3-2*x2+(p-E*(E+2))*x+E*p Setting that polynomial equal to zero gives roots that agree with Euler's equation. In the exceptional groups: ( down to two integer variables) p=16*m ; m-> {1,3,15} E=6*n ; n->{1,2,5} The program works to produce the right roots for {-E,V,F}

			Program to get roots for tetrahedron, (cube, octahedron),
(dodecahedron,
icosahedron):
a = {1, 2, 5}
b = {1, 3, 15}
g[n_, m_] := x /. Solve[e [a[[m]]]*p[b[[m]]] - e [a[[m]]]*(e[a[[
m]]] + 2)*x + p[b[[m]]]* x - 2* x^2 + x^3 == 0, x][[n]]
Table[g[n, m], {n, 1, 3}, {m, 1, 3}]
{{-6, -12, -30}, {4, 6, 12}, {4, 8, 20}}

Cf. edge: A008458; vertex: A118081.

ExpandAll[(x - v)*(x - f)*(x + e)]; e[n_] := 6*n; p[m_] := 16*m; a0 = Table[If[IntegerQ[x /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x +p[p0]* x - 2* x^2 + x^3 == 0, x][[1]]] && IntegerQ[x /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x + p[p0]* x - 2* x^2 + x^3 == 0,x][[2]]] && IntegerQ[x /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x +p[p0]* x - 2* x^2 + x^3 == 0, x][[3]]], {Abs[x] /. Solve[e [m]*p[p0] - e [m]*(e[m] + 2)*x + p[p0]* x - 2* x^2 + x^3 == 0, x][[3]]}, {}], {m, 1, 12}, {p0, 1, 33}]

F roots such that:x^3+(E-V-F)*x^2+(V*F-E(V+F))*x=E*F*V and that are exceptional like ( tetrahedron, cube, octahedron, dodecahedron, icosahedron)

A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 18, 12, 2, 0, 1, 50, 66, 18, 2, 0, 1, 140, 330, 146, 24, 2, 0, 1, 392, 1610, 1070, 258, 30, 2, 0, 1, 1106, 7742, 7580, 2500, 402, 36, 2, 0, 1, 3138, 37058, 52556, 23330, 4850, 578, 42, 2, 0, 1, 8952, 177186, 360402, 212436, 56252, 8350, 786

Offset: 0

R. J. Mathar, Mar 02 2010

The values are computed starting with an auxiliary array which places the centered trinomial numbers A002426, the centered pentanomial numbers A005191, the centered 7-nomial numbers A025012 etc. into separate columns:
.1,....1,......1,.......1,........1,........1,.........1,.........1,.........1
.1,....3,......5,.......7,........9,.......11,........13,........15,........17
.1,....7,.....19,......37,.......61,.......91,.......127,.......169,.......217
.1,...19,.....85,.....231,......489,......891,......1469,......2255,......3281
.1,...51,....381,....1451,.....3951,.....8801,.....17151,.....30381,.....50101
.1,..141,...1751,....9331,....32661,....88913,....204763,....418503,....782153
.1,..393,...8135,...60691,...273127,...908755,...2473325,...5832765,..12354469
.1,.1107,..38165,..398567,..2306025,..9377467,..30162301,..82073295,.197018321
.1,.3139,.180325,.2636263,.19610233,.97464799,.370487485,1163205475,3164588407
This is a subarray of A077042. Rows are A005408, A003215, A063496, A083669 (see A077044) etc. The array A(k,n) is the first differences along each row of this auxiliary array.

			A(k,n) starts in row k=0, column n=0 as:
1,....0,......0,.......0,........0,........0,.........0,.........0,.........0
1,....2,......2,.......2,........2,........2,.........2,.........2,.........2
1,....6,.....12,......18,.......24,.......30,........36,........42,........48
1,...18,.....66,.....146,......258,......402,.......578,.......786,......1026
1,...50,....330,....1070,.....2500,.....4850,......8350,.....13230,.....19720
1,..140,...1610,....7580,....23330,....56252,....115850,....213740,....363650
1,..392,...7742,...52556,...212436,...635628,...1564570,...3359440,...6521704
1,.1106,..37058,..360402,..1907458,..7071442,..20784834,..51910994,.114945026
1,.3138,.177186,.2455938,.16973970,.77854566,.273022686,.792717990,2001382932

R. J. Mathar, Point counts of D_k and some A_k and E_K integer.., arXiv:1002.3844 [math .GT], variable A_k^s(n).

Cf. A008458 (row k=2), A010006 (row k=3), A110907.

cp's OEIS Frontend

A276158 Triangle read by rows: T(n,k) = 6*k*(n + 1 - k) for 0 < k <= n; for k = 0, T(n,0) = n + 1.

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Magma

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A322041 Triangle read by rows: let E denote the standard triangular 6-valent grid in the plane, regarded as a graph with the Eisenstein integers as vertices; row n gives the coordination sequence of the quotient graph E/nE.

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A354958 Coordination sequence for the Manhattan hexagonal lattice with respect to the point (X=0, Y=1).

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PARI

A242941 a(n) is the number of convex uniform tessellations in dimension n.

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A327893 Minesweeper sequence of positive integers arranged in a hexagonal spiral.

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Mathematica

A086461 Symmetric version of square array A086460.

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A130702 Possible sides in the Euler V=E-F+2 as roots in a cubic polynomial of the form: P(x)=(x-V)*(x-F)*(x+E) =x^3+(E-V-F)*x^2+(V*F-E(V+F))*x=E*F*V Solves here for F ( Face, Edge, Vertex).

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A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

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A276158 Triangle read by rows: T(n,k) = 6k(n + 1 - k) for 0 < k <= n; for k = 0, T(n,0) = n + 1.

A130702 Possible sides in the Euler V=E-F+2 as roots in a cubic polynomial of the form: P(x)=(x-V)(x-F)(x+E) =x^3+(E-V-F)x^2+(VF-E(V+F))x=EF*V Solves here for F ( Face, Edge, Vertex).