A008458 -id:A008458 - cp's OEIS frontend

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

1, 3, 12, 9, 24, 15, 36, 21, 48, 27, 60, 33, 72, 39, 84, 45, 96, 51, 108, 57, 120, 63, 132, 69, 144, 75, 156, 81, 168, 87, 180, 93, 192, 99, 204, 105, 216, 111, 228, 117, 240, 123, 252, 129, 264, 135, 276, 141, 288, 147, 300, 153, 312, 159, 324, 165, 336, 171, 348, 177, 360, 183, 372, 189, 384, 195

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the kgd net.

This is one of the Laves tilings.

Robert Israel, Table of n, a(n) for n = 0..10000
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008579, A135556 (partial sums), A298026 (trivalent point).
If the initial 1 is changed to 0 we get A165988 (but we need both sequences, just as we have both A008574 and A008586).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

f3:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 6*n else 3*n; fi; end;
[seq(f3(n),n=0..80)];

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {3, 12, 9, 24}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

a(0)=1; a(2*k) = 12*k, a(2*k+1) = 6*k+3.
G.f.: 1 + 3*x*(x^2+4*x+1)/(1-x^2)^2. - Robert Israel, Jan 21 2018
a(n) = 3*A022998(n), n>0. - R. J. Mathar, Jan 29 2018

A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

Original entry on oeis.org

1, 3, 6, 12, 18, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489, 501, 507, 519, 525, 537, 543, 555

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the deltoidal trihexagonal tiling, or the mta net.
In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020
This is one of the Laves tilings.

Colin Barker, Table of n, a(n) for n = 0..1000
Robert Ferreol, Pavage de diane et son dual
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.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Wikipedia, Laves tilings
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A008574, A298030 (partial sums), A298031 (for a tetravalent node), A298033 (hexavalent node), A306771.

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Join[{1, 3, 6, 12, 18}, LinearRecurrence[{1, 1, -1}, {33, 39, 51}, 60]] (* Jean-François Alcover, Jan 07 2019 *)
Join[{1,3,6,12,18},Table[If[EvenQ[n],9n-15,9n-12],{n,5,70}]] (* Harvey P. Dale, Aug 25 2019 *)

Vec((1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 9*x^5 - 3*x^7) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Theorem: For n >= 5, if n is even then a(n) = 9*n-15, otherwise a(n) = 9*n-12. The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. - N. J. A. Sloane, Jan 24 2018
G.f.: -(3*x^7 - 9*x^5 - 3*x^4 - 4*x^3 - 2*x^2 - 2*x - 1)/((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7. - Colin Barker, Jan 25 2018
a(n) = (3/2)*(6*n - (-1)^n - 9) for n>4. - Bruno Berselli, Jan 25 2018
a(n) = 3*A007310(n-1), n>4. - R. J. Mathar, Jan 29 2018

A298031 Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.

Original entry on oeis.org

1, 4, 10, 16, 30, 36, 48, 54, 66, 72, 84, 90, 102, 108, 120, 126, 138, 144, 156, 162, 174, 180, 192, 198, 210, 216, 228, 234, 246, 252, 264, 270, 282, 288, 300, 306, 318, 324, 336, 342, 354, 360, 372, 378, 390, 396, 408, 414, 426, 432, 444, 450, 462, 468, 480, 486, 498, 504, 516, 522, 534, 540

Offset: 0

N. J. A. Sloane, Jan 21 2018; extended with formula, Jan 24 2018

Also known as the mta net.
This is one of the Laves tilings.
In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020

Colin Barker, Table of n, a(n) for n = 0..1000
Robert Ferreol, Pavage de diane et son dual
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.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, The subgraph H shown in one quadrant of the graph of the tiling.
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A008574, A298032 (partial sums), A298029 (for a trivalent node), A298033 (hexavalent node).

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

f4:=proc(n) local L; L:=[1,4,10,16];
if n<4 then L[n+1] elif (n mod 2) = 0 then 9*n-6 else 9*n-9; fi;
end;
[seq(f4(n),n=0..80)];

Join[{1, 4, 10, 16}, LinearRecurrence[{1, 1, -1}, {30, 36, 48}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Vec((1 + 3*x + 5*x^2 + 3*x^3 + 8*x^4 - 2*x^6) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Theorem: For n >= 4, a(n) = 9*n-6 if n is even, otherwise a(n) = 9*n-9.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: -(2*x^6 - 8*x^4 - 3*x^3 - 5*x^2 - 3*x - 1) / ((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4. - Colin Barker, Jan 25 2018
a(n) = 6*A007494(n-1), n>3. - R. J. Mathar, Jan 29 2018

A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.

Original entry on oeis.org

1, 6, 12, 24, 30, 42, 48, 60, 66, 78, 84, 96, 102, 114, 120, 132, 138, 150, 156, 168, 174, 186, 192, 204, 210, 222, 228, 240, 246, 258, 264, 276, 282, 294, 300, 312, 318, 330, 336, 348, 354, 366, 372, 384, 390, 402, 408, 420, 426, 438, 444, 456, 462, 474, 480, 492, 498, 510, 516, 528, 534, 546, 552

Offset: 0

N. J. A. Sloane, Jan 21 2018, corrected Jan 24 2018

Also known as the mta net.

This is one of the Laves tilings.

Colin Barker, Table of n, a(n) for n = 0..1000
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.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, The subgraph H shown in one 60-degree sector of the graph of the tiling.
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Cf. A008574, A038764 (partial sums), A298029 (coordination sequence for a trivalent node), A298031 (coordination sequence for a tetravalent node).

f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 9*n-6 else 9*n-3; fi; end;
[seq(f6(n),n=0..80)];

Join[{1}, LinearRecurrence[{1, 1, -1}, {6, 12, 24}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

apply( {A298033(n)=if(n,n*3\/2*6-6,1)}, [0..66]) \\ M. F. Hasler, Jan 11 2022

Theorem: For n>0, a(n) = 9*n-6 if n is even, a(n) = 9*n-3 if n is odd.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)*(1 - x^2)).
First differences are 1, 5, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, ...
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - Colin Barker, Jan 25 2018
a(n) = 6*floor((3n-1)/2) for n > 0. - M. F. Hasler, Jan 11 2022

A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 25, 19, 1, 1, 25, 37, 37, 25, 1, 1, 31, 49, 55, 49, 31, 1, 1, 37, 61, 73, 73, 61, 37, 1, 1, 43, 73, 91, 97, 91, 73, 43, 1, 1, 49, 85, 109, 121, 121, 109, 85, 49, 1, 1, 55, 97, 127, 145, 151, 145, 127, 97, 55, 1, 1, 61, 109, 145, 169, 181, 181, 169, 145, 109, 61, 1

Offset: 0

Kolosov Petro, Aug 31 2017

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(1, n, k), i.e T(n, k) is partial case of L(m, n, k) for m = 1.
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
  ----------------------------------------
  k=    0   1   2   3   4   5   6   7   8
  ----------------------------------------
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  7,  1;
  n=3:  1, 13, 13,  1;
  n=4:  1, 19, 25, 19,  1;
  n=5:  1, 25, 37, 37, 25,  1;
  n=6:  1, 31, 49, 55, 49, 31,  1;
  n=7:  1, 37, 61, 73, 73, 61, 37,  1;
  n=8:  1, 43, 73, 91, 97, 91, 73, 43,  1;

Georg Fischer, Table of n, a(n) for n = 0..495 [rows 0..10 and 12..30 from Kolosov Petro]
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Columns k=0..6 give A000012, A016921, A017533, A161705, A103214, A128470, A158065.
Column sums k=0..4 give A000027, A000567, A051866, A051872, A255185.
Row sums give A001093.
Various cases of L(m, n, k): This sequence (m=1), A300656(m=2), A300785(m=3). See comments for L(m, n, k).
Differences of cubes n^3 are T(A000124(n), 1).
Cf. A000124, A000578, A003154, A003215, A007318, A008458, A016969.
Cf. A038593, A055012, A068601, A077028, A094053, A166873, A227776.
Cf. A294317, A302971, A304042.

Flat(List([0..11],n->List([0..n],k->6*k*(n-k)+1))); # Muniru A Asiru, Oct 09 2018

/* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 26 2018

T := (n, k) -> 6*k*(n-k) + 1:
seq(seq(T(n, k), k=0..n), n=0..11); # Muniru A Asiru, Oct 09 2018

T[n_, k_] := 6 k (n - k) + 1; Column[Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jun 02 2019 *)

t(n, k) = 6*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */
trianglerows(9) \\ Felix Fröhlich, Jan 09 2018

def A287326(n,k): return 6*k*(n-k) + 1
flatten([[A287326(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 25 2024

T(n, k) = 6*k*(n-k) + 1.
G.f. of column k: n^k*(1+(6*k-1)*n)/(1-n)^2.
G.f.: (1 - x - x*y + 7*x^2*y)/((1 - x)^2*(1 - x*y)^2). - Stefano Spezia, Oct 09 2018 [Adapted by Stefano Spezia, Sep 25 2024]
From Kolosov Petro, Jun 05 2019: (Start)
T(n, k) = 1/2 * T(A294317(n, k), k) + 1/2.
T(n+1, k) = 2*T(n, k) - T(n-1, k), for n >= k.
T(n, k) = 6*A077028(n, k) - 5.
T(2n, n) = A227776(n).
T(2n+1, n) = A003154(n+1).
T(2n+3, n) = A166873(n+1).
Sum_{k=0..n-1} T(n, k) = Sum_{k=1..n} T(n, k) = A000578(n).
Sum_{k=1..n-1} T(n, k) = A068601(n).
(n+1)^3 - n^3 = T(A000124(n), 1). (End)
Sum_{k=0..n} (-1)^k*T(n, k) = (-1/2)*(1 + (-1)^n)*A016969(floor(n/2) - 1). - G. C. Greubel, Sep 25 2024

A227776 a(n) = 6*n^2 + 1.

Original entry on oeis.org

1, 7, 25, 55, 97, 151, 217, 295, 385, 487, 601, 727, 865, 1015, 1177, 1351, 1537, 1735, 1945, 2167, 2401, 2647, 2905, 3175, 3457, 3751, 4057, 4375, 4705, 5047, 5401, 5767, 6145, 6535, 6937, 7351, 7777, 8215, 8665, 9127, 9601, 10087, 10585, 11095, 11617, 12151

Offset: 0

Clark Kimberling, Jul 30 2013

Least splitter is defined for x < y at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Conjecture: a(n) is the least splitter of s(n) and s(n+1), where s(n) = n*sin(1/n).

			The first eight least splitting rationals for {n*sin(1/n), n >=1 } are these fractions: 6/7, 24/25, 54/55, 96/97, 150/151, 216/217, 294/295, 384/385.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Hexagonal Star Rays.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A227631, A287326.

Cf. A003154, A008458, A003215, A000217, A028896, A054554, A046092, A080855, A045943, A172043, A002378, A033581.

z = 40; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = n*Sin[1/n]; t = Table[r[s[n], s[n + 1]], {n, 1, z}] (* least splitting rationals *); fd = Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)
Array[6 #^2 + 1 &, 45] (* Michael De Vlieger, Nov 08 2017 *)
LinearRecurrence[{3,-3,1},{7,25,55},50] (* Harvey P. Dale, Dec 16 2017 *)

a(n)=6*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (1 + 4*x + 7*x^2)/(1 - x)^3.
a(n) = A287326(2n, n). - Kolosov Petro, Nov 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(6))*coth(Pi/sqrt(6)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(6))*csch(Pi/sqrt(6)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(6))*sinh(Pi/sqrt(3)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(6))*csch(Pi/sqrt(6)).(End)
From Leo Tavares, Nov 20 2021: (Start)
a(n) = A003154(n+1) - A008458(n). See Hexagonal Star Rays illustration.
a(n) = A003215(n) + A028896(n-1).
a(n) = A054554(n+1) + A046092(n).
a(n) = A080855(n) + A045943(n).
a(n) = A172043(n) + A002378(n).
a(n) = A033581(n) + 1. (End)
E.g.f.: exp(x)*(1 + 6*x + 6*x^2). - Stefano Spezia, Sep 14 2024

a(0) = 1 prepended by Robert P. P. McKone, Oct 09 2023

A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

Original entry on oeis.org

1, 12, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (1).

For other sequences based upon MagicNKZ(n,k,z):
..... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7
---------------------------------------------------------------------------
z = 0 | A000007 | A019590 | .......MagicNKZ(n,k,0) = A008292(n,k+1) .......
z = 1 | A000012 | A040000 | A101101 | thisSeq | A101100 | ....... | .......
z = 2 | A000027 | A005408 | A008458 | A101103 | A101095 | ....... | .......
z = 3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | .......
z = 4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | .......
z = 5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181
z = 6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177
z = 7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523
z = 8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015
z = 9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541
Cf. A101095 for an expanded table and more about MagicNKZ.

MagicNKZ = Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 4, 4}, {z, 1, 1}, {k, 0, 34}]
Join[{1, 12, 23},LinearRecurrence[{1},{24},56]] (* Ray Chandler, Sep 23 2015 *)

a(k) = MagicNKZ(4,k,1) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n (cf. A101095). That is, a(k) = Sum_{j=0..k+1} (-1)^j*binomial(4, j)*(k-j+1)^4.
a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4. - Joerg Arndt, Nov 30 2014
G.f.: x*(1+11*x+11*x^2+x^3)/(1-x). - Colin Barker, Apr 16 2012

New name from Joerg Arndt, Nov 30 2014

Original Formula edited and Crossrefs table added by Danny Rorabaugh, Apr 22 2015

A101095 Fourth difference of fifth powers (A000584).

Original entry on oeis.org

1, 28, 121, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.

The (Worpitzky/Euler/Pascal Cube) "MagicNKZ" algorithm is: MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n, with k>=0, n>=1, z>=0. MagicNKZ is used to generate the n-th accumulation sequence of the z-th row of the Euler Triangle (A008292). For example, MagicNKZ(3,k,0) is the 3rd row of the Euler Triangle (followed by zeros) and MagicNKZ(10,k,1) is the partial sums of the 10th row of the Euler Triangle. This sequence is MagicNKZ(5,k-1,2).

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive Machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Fourth differences of A000584, third differences of A022521, second differences of A101098, and first differences of A101096.
For other sequences based upon MagicNKZ(n,k,z):
...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8
--------------------------------------------------------------------------------------
z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
z =  2 | A000027 | A005408 | A008458 | A101103 | thisSeq | ....... | ....... | .......
z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | .......
z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | .......
z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | .......
z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182
z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178
z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524
z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016
z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542
z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636
z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642
z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647
z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......
--------------------------------------------------------------------------------------
Cf. A047969.

I:=[1,28,121,240,360]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, May 07 2015

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 2, 2}, {k, 0, 34}]
CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)
Join[{1,28,121,240},Differences[Range[50]^5,4]] (* or *) LinearRecurrence[{2,-1},{1,28,121,240,360},50] (* Harvey P. Dale, Jun 11 2016 *)

a(n)=if(n>3, 120*n-240, 33*n^2-72*n+40) \\ Charles R Greathouse IV, Oct 11 2015

[1,28,121]+[120*(k-2) for k in range(4,36)] # Danny Rorabaugh, Apr 23 2015

a(k+1) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n; n = 5, z = 2.
For k>3, a(k) = Sum_{j=0..4} (-1)^j*binomial(4, j)*(k - j)^5 = 120*(k - 2).
a(n) = 2*a(n-1) - a(n-2), n>5. G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (1-x)^2. - Colin Barker, Mar 01 2012

MagicNKZ material edited, Crossrefs table added, SeriesAtLevelR material removed by Danny Rorabaugh, Apr 23 2015

Name changed and keyword 'uned' removed by Danny Rorabaugh, May 06 2015

A110907 Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n.

Original entry on oeis.org

1, 12, 50, 108, 194, 300, 434, 588, 770, 972, 1202, 1452, 1730, 2028, 2354, 2700, 3074, 3468, 3890, 4332, 4802, 5292, 5810, 6348, 6914, 7500, 8114, 8748, 9410, 10092, 10802, 11532, 12290, 13068, 13874, 14700, 15554, 16428, 17330, 18252, 19202

Offset: 0

N. J. A. Sloane, Apr 15 2008

This lattice consists of all points (x,y,z) where x,y,z are integers with an even sum.
The L_infinity norm of a vector is the largest component in absolute value.
The sequence for the D_k lattice has the terms ((2*n+1)^k-(2*n-1)^k)/2, if k is even, and the terms ((2n+1)^k-(2*n-1)^k)/2+(-1)^n if k is odd (like here for k=3). The sequence for A_2 is A008458, for A_3 A010006, for A_4 the first differences of A083669. A_5 is 2+2*n^2*(25+44*n^2) if n>0, and 1 if n=0. - R. J. Mathar, Feb 09 2010

			a(0) = 1: 000
a(1) = 12: +-1 +-1 0, where the 0 can be in any of the three coordinates
a(2) = 50: +-2 0 0 (6), +-2 +-1 +-1 (24), +-2 +-2 0 (12), +-2 +-2 +-2 (8).

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.

R. J. Mathar, Point counts of D_k and some A_k and E_k integer lattices inside hypercubes arXiv:1002.3844 [math.GT], 2010.
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to f.c.c. lattice
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Cf. A117216, A022144, A010014, A175112 (D_5), A175114 (D_6).

A110907 := proc(n) a :=0 ; for x from -n to n do for y from -n to n do for z from -n to n do if type(x+y+z,'even') then m := max( abs(x),abs(y),abs(z)) ; if m = n then a := a+1 ; end if; end if; end do ; end do ; end do ; a ; end proc: seq(A110907(n),n=0..40) ; # R. J. Mathar, Feb 03 2010

a[0] = 1; a[n_] := 1 + (-1)^n + 12*n^2;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 16 2017, after R. J. Mathar *)

From R. J. Mathar, Feb 03 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>4.
a(n) = 1 + (-1)^n + 12*n^2, n>0.
G.f.: 1 - 2*x*(6 + 13*x + 4*x^2 + x^3)/((1+x)*(x-1)^3). (End)

I would like to get analogous sequences for A_2, A_4, A_5, ..., D_4 (see A117216), D_5, ..., E_6, E_7, E_8.
Extended by R. J. Mathar, Feb 03 2010
Removed the "conjectured" attribute from formulas - R. J. Mathar, Feb 27 2010

A101101 a(1)=1, a(2)=5, and a(n)=6 for n >= 3.

Original entry on oeis.org

1, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Previous name was: The first summation of row 3 of Euler's triangle - a row that will recursively accumulate to the power of 3.

Decimal expansion of 47/30. - Elmo R. Oliveira, Aug 09 2024

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al.), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [broken link: domain now owned by a domain grabber]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883.
Eric Weisstein, Link to section of MathWorld: Eulerian Number.
Eric Weisstein, Link to section of MathWorld: Nexus number.
Eric Weisstein, Link to section of MathWorld: Finite Differences.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Within the "cube" of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward:
Before: this_sequence, A008458, A003215, A000578, A000537, A024166 or A024166, A101094, A101097, A101102.
Above: this_sequence, below: A101104, A101100.
Within the "cube" of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward and r backward:
Above: this_sequence, below: A101103, A101096.

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 3, 3}, {z, 1, 1}, {k, 0, 34}] (* OR *)
SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 3, 3}, {r, -3, -3}, {x, 4, 35}]
Join[{1, 5},LinearRecurrence[{1},{6},78]] (* Ray Chandler, Sep 23 2015 *)

G.f.: x*(1+4*x+x^2)/(1-x). - L. Edson Jeffery, Jan 29 2012

I wish the sequence was as interesting as the list of references! - N. J. A. Sloane

New name from Joerg Arndt, Nov 30 2014

cp's OEIS Frontend

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

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A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

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A298031 Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.

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A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.

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A287326 Triangle read by rows: T(n, k) = 6*k*(n-k) + 1; n >= 0, 0 <= k <= n.

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A227776 a(n) = 6*n^2 + 1.

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A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

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A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.