A008412 -id:A008412 - cp's OEIS frontend

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

0, 2, 16, 66, 192, 450, 912, 1666, 2816, 4482, 6800, 9922, 14016, 19266, 25872, 34050, 44032, 56066, 70416, 87362, 107200, 130242, 156816, 187266, 221952, 261250, 305552, 355266, 410816, 472642, 541200, 616962, 700416, 792066

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A035596, A035597, A035599, A035600, A035601, A035602, A035603, A035604, A035605, A035606.
Cf. A001845, A001846, A008412, A014820.
Column 4 of A035607, A266213, A343599.
Row 4 of A113413, A119800, A122542.

[( 2*n^4 +4*n^2 )/3: n in [0..40]]; // Vincenzo Librandi, Apr 22 2012

f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm

CoefficientList[Series[2*x*(1+x)^3/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,16,66,192},50] (* Harvey P. Dale, Dec 11 2019 *)

a(n)=2*n^2*(n^2+2)/3 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = 2*n^2*(n^2 + 2)/3. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^3/(1-x)^5. - Colin Barker, Apr 15 2012
a(n) = 2*A014820(n-1). - R. J. Mathar, Dec 10 2013
a(n) = a(n-1) + A035597(n) + A035597(n-1). - Bruce J. Nicholson, Mar 11 2018
From Shel Kaphan, Feb 28 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=4.
a(n) = A001846(n) - A001845(n).
a(n) = A008412(n)*n/4. (End)
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 - 3*Pi*coth(sqrt(2)*Pi)/(8*sqrt(2)) + 3/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/16 + 3*Pi*cosech(sqrt(2)*Pi)/(8*sqrt(2)) - 3/16. (End)
E.g.f.: 2*exp(x)*x*(3 + 9*x + 6*x^2 + x^3)/3. - Stefano Spezia, Mar 14 2024

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

A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 9, 42, 140, 383, 925, 2056, 4316, 8705, 17069, 32810, 62192, 116743, 217673, 404000, 747496, 1380177, 2544865, 4688186, 8631620, 15886111, 29230725, 53776968, 98926372, 181971057, 334716197, 615660634, 1132400520

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn15 and Kn25 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-3,3,-3,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=29: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 od: seq(a(n),n=0..nmax);

RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+(8n^3-48n^2+112n-96)/3},a,{n,30}] (* or *) LinearRecurrence[{5,-9,7,-3,3,-3,1},{0,0,1,9,42,140,383},30] (* Harvey P. Dale, Dec 04 2019 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+7)-(12+4*n+4*n^2) with a(0)=0.
a(n) = sum(A008412(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+4,k+4),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^4)/((1-x)^4*(1-x-x^2-x^3)).

A217873 a(n) = 4n(n^2 + 2)/3.

Original entry on oeis.org

0, 4, 16, 44, 96, 180, 304, 476, 704, 996, 1360, 1804, 2336, 2964, 3696, 4540, 5504, 6596, 7824, 9196, 10720, 12404, 14256, 16284, 18496, 20900, 23504, 26316, 29344, 32596, 36080, 39804, 43776, 48004, 52496, 57260, 62304, 67636, 73264, 79196, 85440, 92004

Offset: 0

M. F. Hasler, Oct 13 2012

Occurs as 4th column in the table A142978 of figurate numbers for n-dimensional cross polytope.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006527, A008412, A008590, A022144, A082138, A112087, A142978, A174794, A292022.

[4*n*(n^2+2)/3: n in [0..45]]; // Vincenzo Librandi, Nov 08 2012

Table[4n(n^2 + 2)/3, {n, 0, 39}] (* Alonso del Arte, Oct 22 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,4,16,44},50] (* Harvey P. Dale, Mar 16 2015 *)

makelist(4*n*(n^2+2)/3, n, 0, 41); /* Martin Ettl, Oct 15 2012 */

PARI
```
a(n)=(n^2+2)*n/3*4
```

a(n) = 4*A006527(n).
From Peter Luschny, Oct 14 2012: (Start)
a(n) = A008412(n)/2.
a(n) = A174794(n+1) - 1.
First differences are in A112087.
Second differences are in A008590 and A022144.
Binomial transformation of (a(n), n > 0) is A082138.  (End)
G.f.: 4*x*(1 + x^2)/(x - 1)^4. - R. J. Mathar, Oct 15 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=4, a(2)=16, a(3)=44. - Harvey P. Dale, Mar 16 2015
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*exp(x)*x*(3 + 3*x + x^2)/3.
a(n) = A292022(n)/3. (End)

A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 4, 1, 1, 0, 4, 2, 1, 10, 4, 5, 3, 1, 0, 10, 8, 7, 4, 1, 20, 10, 14, 13, 10, 5, 1, 0, 20, 20, 22, 20, 14, 6, 1, 35, 20, 30, 34, 35, 30, 19, 7, 1, 0, 35, 40, 50, 56, 55, 44, 25, 8, 1, 56, 35, 55, 70, 84, 91, 85, 63, 32, 9, 1, 0, 56, 70, 95, 120, 140, 146, 129, 88, 40, 10, 1

Offset: 0

Henry Bottomley, Nov 20 2000

			Rows are (1,0,4,0,10,0,20,...), (1,1,4,4,10,10,20,...), (1,2,5,8,14,20,30,...), (1,3,7,13,22,34,50,...), (1,4,10,20,35,56,84,...) etc.

Rows are A000292 with zeros, A058187 (A000292 with terms duplicated), A006918, A002623, A000292, A000330, A005900, A001845, A008412.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(4, 1)=4, T(0, 2n)=T(4, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^4.

A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 4, 1, 6, 18, 1, 8, 32, 88, 1, 10, 50, 170, 450, 1, 12, 72, 292, 912, 2364, 1, 14, 98, 462, 1666, 4942, 12642, 1, 16, 128, 688, 2816, 9424, 27008, 68464, 1, 18, 162, 978, 4482, 16722, 53154, 148626, 374274, 1, 20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, 1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898

Offset: 0

R. J. Mathar, Apr 21 2021

			The full array starts
     1      2      2      2      2      2      2      2      2
     1      4      8     12     16     20     24     28     32
     1      6     18     38     66    102    146    198    258
     1      8     32     88    192    360    608    952   1408
     1     10     50    170    450   1002   1970   3530   5890
     1     12     72    292    912   2364   5336  10836  20256
     1     14     98    462   1666   4942  12642  28814  59906
     1     16    128    688   2816   9424  27008  68464 157184
     1     18    162    978   4482  16722  53154 148626 374274

J. Schroder, Generalized Schroder Numbers and the Rotation principle, J. Int. Seq. 10 (2007) # 07.7.7, Theorem 4.2.

Cf. A035607 (by antidiags), A008574 (n=1), A005899 (n=2), A008412 (n=3), A008413 (n=4), A008414 (n=5), A001105 (k=2), A035597 (k=3), A035598 (k=4).

Main diagonal gives A050146(n+1).

A343599 := proc(n,k)
    local g,x,y ;
    g := (1+y)/(1-x-y-x*y) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,k) ;
end proc:

T[n_, k_] := Module[{x, y}, SeriesCoefficient[(1 + y)/(1 - x - y - x*y), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 16 2023 *)

G.f.: (1+y)/(1-x-y-x*y).

T(n,k) = A008288(n,k) + A008288(n,k-1).

A035878 Number of points of l_1 norm n in the "diamond" lattice D^+_4.

Original entry on oeis.org

1, 0, 40, 32, 272, 160, 888, 448, 2080, 960, 4040, 1760, 6960, 2912, 11032, 4480, 16448, 6528, 23400, 9120, 32080, 12320, 42680, 16192, 55392, 20800, 70408, 26208, 87920, 32480, 108120, 39680, 131200, 47872, 157352, 57120, 186768, 67488, 219640, 79040, 256160

Offset: 0

Joan Serra-Sagrista (jserra(AT)ccd.uab.es)

			This 4D lattice consists of points with coordinates that have even sum and are either all integer or all half-integer. (It is actually similar to Z^4.) The a(2) = 40 lattice vectors having l_1 norm 2 include: +-(1,1,1,1)/2, 6 permutations of (1,1,-1,-1)/2, 6 permutations with 4 choices of signs in (+-1,+-1,0,0), and 4 permutations with 2 choices of signs in (+-2,0,0,0), totaling 2 + 6 + 6*4 + 4*2 = 40.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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-Sagristà, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A035877, A035879, A010369, A035881-A035926, A010006, A008412, A005925.

n := 4; A035878 := proc(m) global n; local k,t1; t1 := 2^(n-1)*binomial((n+2*m)/2-1,n-1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(n,k)*binomial(m-1,k-1),k=0..n); fi; t1; end;

f[m_, n_] := 2^(n-1) *Binomial[(n + 2*m)/2 - 1, n - 1] + If[EvenQ[m], 2 *n* Hypergeometric2F1[1-m, 1-n, 2, 2], 0]; f[0, ] = 1; Table[f[m, 4], {m, 0, 32}] (* _Jean-François Alcover, Apr 18 2013, after Maple *)
CoefficientList[Series[(x^8 + 36 x^6 + 32 x^5 + 118 x^4 + 32 x^3 + 36 x^2 + 1)/((x - 1)^4 (x + 1)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2013 *)

For n>0, a(n) = ( 2n^2 + 1 + (n^2+2)*(-1)^n ) * 4n/3.

G.f.: (x^8+36*x^6+32*x^5+118*x^4+32*x^3+36*x^2+1) / ((x-1)^4*(x+1)^4). - Colin Barker, Nov 18 2012

Recomputed by N. J. A. Sloane, Nov 27 1998
More terms from Vincenzo Librandi, Oct 21 2013
Name edited by Andrey Zabolotskiy, Aug 29 2022

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).

A236967 Expansion of (1+3x)^2/(1-3x)^2.

Original entry on oeis.org

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Index entries for linear recurrences with constant coefficients, signature (6,-9)

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 |         |         |         |         |
k = 3 | A099856 | A236967 |         |         |         |         |
k = 4 | A081654 |         |         |         |         |         |
-------------------------------------------------------------------

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Edited by Wolfdieter Lang, May 07 2014

cp's OEIS Frontend

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

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A008416 Coordination sequence for 8-dimensional cubic lattice.

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A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

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A217873 a(n) = 4*n*(n^2 + 2)/3.

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A057884 A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.

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A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

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Maple

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A035878 Number of points of l_1 norm n in the "diamond" lattice D^+_4.

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

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A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

A217873 a(n) = 4n(n^2 + 2)/3.

A236967 Expansion of (1+3x)^2/(1-3x)^2.