A008415 -id:A008415 - cp's OEIS frontend

A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

4, 8, 6, 12, 18, 8, 16, 38, 32, 10, 20, 66, 88, 50, 12, 24, 102, 192, 170, 72, 14, 28, 146, 360, 450, 292, 98, 16, 32, 198, 608, 1002, 912, 462, 128, 18, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 40, 326, 1408, 3530, 5336, 4942, 2816, 978, 200, 22

Offset: 1

Thomas Wieder, Jul 30 2006, Aug 06 2006

			The second row of the table is: 6, 18, 38, 66, 102, 146, 198, 258, 326, ... = A005899 = number of points on surface of octahedron.
The third column of the table is: 12, 38, 88, 170, 292, 462, 688, 978, 1340, ... = A035597 = number of points of L1 norm 3 in cubic lattice Z^n.
The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420.
The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603.
The main diagonal seems to be A050146.
Square array A(n,k) begins:
   4,   8,   12,   16,    20,    24,     28,     32,      36, ...
   6,  18,   38,   66,   102,   146,    198,    258,     326, ...
   8,  32,   88,  192,   360,   608,    952,   1408,    1992, ...
  10,  50,  170,  450,  1002,  1970,   3530,   5890,    9290, ...
  12,  72,  292,  912,  2364,  5336,  10836,  20256,   35436, ...
  14,  98,  462, 1666,  4942, 12642,  28814,  59906,  115598, ...
  16, 128,  688, 2816,  9424, 27008,  68464, 157184,  332688, ...
  18, 162,  978, 4482, 16722, 53154, 148626, 374274,  864146, ...
  20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Other versions: A035607, A113413, A122542, A266213.

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146.

A:= proc(m, n)  option remember;
      `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # Alois P. Heinz, Apr 21 2012

A[m_, n_] := A[m, n] = If[n == 0, 1, If[m == 0, 2, A[m, n-1] + A[m-1, n] + A[m-1, n-1]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

A(m,n) = A(m,n-1) + A(m-1,n) + A(m-1,n-1), A(m,0)=1, A(0,0)=1, A(0,n)=2.

Offset and typos corrected by Alois P. Heinz, Apr 21 2012

A008416 Coordination sequence for 8-dimensional cubic lattice.

Original entry on oeis.org

1, 16, 128, 688, 2816, 9424, 27008, 68464, 157184, 332688, 658048, 1229360, 2187520, 3732560, 6140800, 9785072, 15158272, 22900496, 33830016, 48978352, 69629696, 97364944, 134110592, 182192752, 244396544, 324031120, 425000576

Offset: 0

N. J. A. Sloane

Coordination sequence for 8-dimensional cyclotomic lattice Z[zeta_16].

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Ross McPhedran, Numerical Investigations of the Keiper-Li Criterion for the Riemann Hypothesis, arXiv:2311.06294 [math.NT], 2023. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008418, A008420.

Cf. A113413, A122542.

CoefficientList[Series[((1 + x)/(1 - x))^8, {x, 0, 26}], x] (* Michael De Vlieger, Dec 18 2017 *)

G.f.: ((1+x)/(1-x))^8.
a(n) = A008415(n) + A008415(n-1) + a(n-1). - Bruce J. Nicholson, Dec 17 2017
n*a(n) = 16*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

A019563 Coordination sequence for C_7 lattice.

Original entry on oeis.org

1, 98, 1666, 12642, 59906, 209762, 596610, 1459810, 3188738, 6376034, 11879042, 20889442, 35011074, 56345954, 87588482, 132127842, 194158594, 278799458, 392220290, 541777250, 736156162, 985524066

Offset: 0

Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A008415.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x+1)*(1+90*x+911*x^2+2092*x^3+911*x^4 + 90*x^5+x^6)/(1-x)^7 )); // G. C. Greubel, Dec 08 2018

seq(coeff(series((x+1)*(1+90*x+911*x^2+2092*x^3+911*x^4+90*x^5+x^6)/(1-x)^7,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 08 2018

Join[{1}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {98, 1666, 12642, 59906, 209762, 596610, 1459810}, 21]] (* Jean-François Alcover, Dec 08 2018 *)

my(x='x+O('x^30)); Vec((x+1)*(1+90*x+911*x^2+2092*x^3+911*x^4 + 90*x^5+x^6)/(1-x)^7) \\ G. C. Greubel, Dec 08 2018

s=((x+1)*(1+90*x+911*x^2+2092*x^3+911*x^4 + 90*x^5+x^6)/(1-x)^7 ).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 08 2018

G.f.: (x + 1)*(1 + 90*x + 911*x^2 + 2092*x^3 + 911*x^4 + 90*x^5 + x^6)/(1 - x)^7.

a(n) = A008415(2*n). - Seiichi Manyama, Jun 08 2018

A191596 Expansion of (1+x)^4/(1-x)^7.

Original entry on oeis.org

1, 11, 62, 242, 743, 1925, 4396, 9108, 17469, 31471, 53834, 88166, 139139, 212681, 316184, 458728, 651321, 907155, 1241878, 1673882, 2224607, 2918861, 3785156, 4856060, 6168565, 7764471, 9690786, 12000142, 14751227, 18009233, 21846320

Offset: 0

Bruno Berselli, Jun 08 2011

The first, second and third differences are in A069038, A001846 and A008412, respectively.
Inverse binomial transform of this sequence: 1, 10, 41, 88, 104, 64, 16, 0, 0 (0 continued).
Also (by Superseeker), the n-th coefficient of the expansion of ((1+x)^4/(1-x)^7)*(1+x)^n is A006976(n-1).

Bruno Berselli, Table of n, a(n) for n = 0..1000
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A008415, A001848, A069039, A008412, A001846, A069038, A061927 (for type of g.f.).

[(2*n^6+18*n^5+80*n^4+210*n^3+323*n^2+267*n+90)/90: n in [0..30]]; // Vincenzo Librandi, Jun 08 2011

A191596:=n->(n+1)*(n+2)*(2*n^4+12*n^3+40*n^2+66*n+45)/90: seq(A191596(n), n=0..40); # Wesley Ivan Hurt, Nov 20 2014

CoefficientList[Series[(1 + x)^4/(1 - x)^7, {x, 0, 30}], x] (* Wesley Ivan Hurt, Nov 20 2014 *)

makelist(coeff(taylor((1+x)^4/(1-x)^7, x, 0, n), x, n), n, 0, 30);

a(n)=(((((n+n+18)*n+80)*n+210)*n+323)*n+267)/90*n+1 \\ Charles R Greathouse IV, Jun 08 2011

G.f.: (1+x)^4/(1-x)^7.
a(n) = (n+1)*(n+2)*(2*n^4+12*n^3+40*n^2+66*n+45)/90.
a(n) = a(-n-3) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
By Superseeker:
a(n)+a(n+1) = A069039(n+2),
a(n+2)-a(n) = A001847(n+2),
a(n+2)+2*a(n+1)+a(n) = A001848(n+2).

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A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

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A008416 Coordination sequence for 8-dimensional cubic lattice.

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A019563 Coordination sequence for C_7 lattice.

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A191596 Expansion of (1+x)^4/(1-x)^7.

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