A008527 -id:A008527 - cp's OEIS frontend

A008383 Coordination sequence for A_4 lattice.

1, 20, 110, 340, 780, 1500, 2570, 4060, 6040, 8580, 11750, 15620, 20260, 25740, 32130, 39500, 47920, 57460, 68190, 80180, 93500, 108220, 124410, 142140, 161480, 182500, 205270, 229860, 256340, 284780, 315250, 347820, 382560, 419540, 458830, 500500, 544620

Offset: 0

N. J. A. Sloane and J. H. Conway

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; 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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005901, A008384, A008385.

[n eq 0 select 1 else 5*n*(7*n^2+5)/3: n in [0..45]]; // G. C. Greubel, May 25 2023

a:= n-> `if`(n=0, 1, 35/3*n^3+25/3*n): seq (a(n), n=0..50);

CoefficientList[Series[(1+16x+36x^2+16x^3+x^4)/(1-x)^4,{x,0,40}],x] (* Harvey P. Dale, Dec 01 2013 *)
Join[{1}, LinearRecurrence[{4, -6, 4, -1}, {20, 110, 340, 780}, 40]] (* Jean-François Alcover, Jan 07 2019 *)

[5*n*(7*n^2+5)/3+int(n==0) for n in range(46)] # G. C. Greubel, May 25 2023

a(n) = 5*n*(7*n^2 + 5)/3, a(0) = 1.
G.f.: (1+16*x+36*x^2+16*x^3+x^4)/(1-x)^4 = 1+10*x*(2+3*x+2*x^2)/(x-1)^4. - Colin Barker, Apr 13 2012
E.g.f.: (1/3)*(3 + 5*x*(12 + 21*x + 7*x^2)*exp(x)). - G. C. Greubel, May 25 2023

A008385 Coordination sequence for A_5 lattice.

Original entry on oeis.org

1, 30, 240, 1010, 2970, 7002, 14240, 26070, 44130, 70310, 106752, 155850, 220250, 302850, 406800, 535502, 692610, 882030, 1107920, 1374690, 1687002, 2049770, 2468160, 2947590, 3493730, 4112502, 4810080, 5592890, 6467610, 7441170, 8520752

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; 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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[n eq 0 select 1 else (21*n^4 +35*n^2 +4)/2: n in [0..50]]; // G. C. Greubel, May 26 2023

Maple
```
1, seq((21*n^4 +35*n^2 +4)/2, n=1..50);
```

Table[n^2*(21*n^2 +35)/2 +2 -Boole[n==0], {n,0,50}] (* G. C. Greubel, May 26 2023 *)

A008385[n]:=21/2*n^4+35/2*n^2+2$
makelist(A008385[n],n,0,30); /* Martin Ettl, Oct 26 2012 */

[n^2*(21*n^2 +35)/2 +2 -int(n==0) for n in range(51)] # G. C. Greubel, May 26 2023

a(n) = (21*n^4 + 35*n^2 + 4)/2, a(0) = 1.
G.f.: (1+x)*(1+24*x+76*x^2+24*x^3+x^4)/(1-x)^5. - Colin Barker, Apr 13 2012
E.g.f.: (1/2)*(-2 + (4 + 56*x + 182*x^2 + 126*x^3 + 21*x^4)*exp(x)). - G. C. Greubel, May 26 2023

A008531 Coordination sequence for {A_4}* lattice.

Original entry on oeis.org

1, 10, 50, 150, 340, 650, 1110, 1750, 2600, 3690, 5050, 6710, 8700, 11050, 13790, 16950, 20560, 24650, 29250, 34390, 40100, 46410, 53350, 60950, 69240, 78250, 88010, 98550, 109900, 122090, 135150, 149110, 164000, 179850, 196690, 214550, 233460, 253450

Offset: 0

N. J. A. Sloane

Coordination sequence for 4-dimensional cyclotomic lattice Z[zeta_10].

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A222408.

Concatenation([1], List([1..45], n-> 5*n*(1+n^2))); # G. C. Greubel, Nov 10 2019

[1] cat [5*n*(1+n^2): n in [1..45]]; // G. C. Greubel, Nov 10 2019

Maple
```
1, seq( 5*k^3+5*k, k=1..40);
```

CoefficientList[Series[(1 +6x +16x^2 +6x^3 +x^4)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,10,50,150,340},40] (* Harvey P. Dale, Jun 09 2016 *)

a(n)=5*n*(n^2+1) \\ Charles R Greathouse IV, Mar 08 2013

[1]+[5*n*(1+n^2) for n in (1..45)] # G. C. Greubel, Nov 10 2019

G.f.: (1 +6*x +16*x^2 +6*x^3 +x^4)/(1-x)^4. - Colin Barker, Sep 21 2012

E.g.f.: 1 + x*(10 + 15*x + 5*x^2)*exp(x). - G. C. Greubel, Nov 10 2019

A008349 Crystal ball sequence for E_8 lattice.

Original entry on oeis.org

1, 241, 9361, 131041, 996001, 5109841, 20015281, 64495681, 179375041, 444798001, 1006201681, 2111519521, 4162485601, 7783236241, 13909734001, 23903867521, 39696408961, 63963339121, 100340378641, 153680892001

Offset: 0

N. J. A. Sloane and J. H. Conway

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, 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).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

[57/7*n^8 + 108/7*n^7 + 30*n^6 + 72*n^5 + 39*n^4 + 36*n^3 + 300/7*n^2 - 24/7*n + 1: n in [0..40]]; // Vincenzo Librandi, Dec 16 2015

57/7*n^8+108/7*n^7+30*n^6+72*n^5+39*n^4+36*n^3+300/7*n^2-24/7*n+1;

CoefficientList[Series[(1+232x+7228x^2+107224x^5+133510x^4+ 55384x^3+ 24508x^6+ 232x^7+ x^8)/(1-x)^9,{x,0,30}],x] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,241,9361,131041,996001,5109841,20015281,64495681,179375041},30] (* Harvey P. Dale, Jun 12 2012 *)

a(n)=(57*n^8 + 108*n^7 + 210*n^6 + 504*n^5 + 273*n^4 + 252*n^3 + 300*n^2 - 24*n + 7)/7 \\ Charles R Greathouse IV, Feb 10 2017

A008349_list, m = [], [328320, -1071360, 1347840, -812160, 233280, -25920, 240, 0, 1]
for _ in range(10**2):
    A008349_list.append(m[-1])
    for i in range(8):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

G.f.: (1 + 232*x + 7228*x^2 + 55384*x^3 + 133510*x^4 + 107224*x^5 + 24508*x^6 + 232*x^7 + x^8)/(1 - x)^9.

a(0)=1, a(1)=241, a(2)=9361, a(3)=131041, a(4)=996001, a(5)=5109841, a(6)=20015281, a(7)=64495681, a(8)=179375041, a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - Harvey P. Dale, Jun 12 2012

The values given by O'Keeffe are incorrect.

A008387 Coordination sequence for A_6 lattice.

Original entry on oeis.org

1, 42, 462, 2562, 9492, 27174, 65226, 137886, 264936, 472626, 794598, 1272810, 1958460, 2912910, 4208610, 5930022, 8174544, 11053434, 14692734, 19234194, 24836196, 31674678, 39944058, 49858158, 61651128, 75578370, 91917462, 110969082

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; 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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row 6 of A103881.

[n eq 0 select 1 else 7*n*(11*n^4+35*n^2+14)/10: n in [0..50]]; // G. C. Greubel, May 26 2023

1, seq(7*n*(11*n^4+35*n^2+14)/10, n=1..40);

LinearRecurrence[{6,-15,20,-15,6,-1}, {1,42,462,2562,9492,27174,65226}, 30] (* Jean-François Alcover, Jan 07 2019 *)

[7*n*(11*n^4 +35*n^2 +14)/10 +int(n==0) for n in range(51)] # G. C. Greubel, May 26 2023

a(n) = S(n,6) = 7*n*(11*n^4 + 35*n^2 + 14)/10, with S(n,m) = Sum_{k=0..m} binomial(m,k)^2 * binomial(n-k+m-1, m-1), for n > 0, and a(0) = 1.
G.f.: (1+36*x+225*x^2+400*x^3+225*x^4+36*x^5+x^6)/(1-x)^6 = 1 + 42*x*(1+5*x+10*x^2+5*x^3+x^4)/(1-x)^6. - Colin Barker, Sep 26 2012
E.g.f.: 1 + (1/10)*x*(420 + 1890*x + 2170*x^2 + 770*x^3 + 77*x^4)*exp(x). - G. C. Greubel, May 26 2023

A008528 Coordination sequence for 4-dimensional RR-centered di-isohexagonal orthogonal lattice.

Original entry on oeis.org

1, 18, 102, 318, 732, 1410, 2418, 3822, 5688, 8082, 11070, 14718, 19092, 24258, 30282, 37230, 45168, 54162, 64278, 75582, 88140, 102018, 117282, 133998, 152232, 172050, 193518, 216702, 241668, 268482, 297210, 327918, 360672, 395538, 432582, 471870, 513468, 557442, 603858, 652782, 704280

Offset: 0

N. J. A. Sloane

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Concatenation([1], List([1..45], n-> n*(7+11*n^2) )); # G. C. Greubel, Nov 09 2019

I:=[1, 18, 102, 318,732]; [n le 5 select I[n] else 4*Self(n-1) -6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 19 2012

Maple
```
1, seq(11*k^3+7*k, k=1..45);
```

CoefficientList[Series[1+6*x*(3+5*x+3*x^2)/(1-x)^4,{x,0,45}],x] (* Vincenzo Librandi, Jun 19 2012 *)
LinearRecurrence[{4,-6,4,-1},{1,18,102,318,732},45] (* Harvey P. Dale, Apr 27 2017 *)

vector(46, n, if(n==1,1,(n-1)*(7+11*(n-1)^2)) ) \\ G. C. Greubel, Nov 09 2019

[1]+[n*(7+11*n^2) for n in (1..45)] # G. C. Greubel, Nov 09 2019

a(n) = n*(11*n^2 + 7) with n>0, with a(0)=1.
G.f.: 1 + 6*x*(3 + 5*x + 3*x^2)/(1-x)^4. - R. J. Mathar, Sep 04 2011
E.g.f.: 1 + x*(18 + 33*x + 11*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A108100 a(n) = (2n-1)^2 + (2n+1)^2.

Original entry on oeis.org

2, 10, 34, 74, 130, 202, 290, 394, 514, 650, 802, 970, 1154, 1354, 1570, 1802, 2050, 2314, 2594, 2890, 3202, 3530, 3874, 4234, 4610, 5002, 5410, 5834, 6274, 6730, 7202, 7690, 8194, 8714, 9250, 9802, 10370, 10954, 11554, 12170, 12802, 13450, 14114, 14794, 15490

Offset: 0

Dorthe Roel (dorthe_roel(AT)hotmail.com or dorthe.roel1(AT)skolekom.dk), Jun 07 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Apart from leading term, same as A008527.

Cf. A000466, A016754, A053755, A158444.

[8*n^2+2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

Table[8n^2+2,{n,0,50}]  (* Harvey P. Dale, Feb 20 2011 *)

makelist(8*n^2+2,n,0,30); /* Martin Ettl, Nov 12 2012 */

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

From R. J. Mathar, Aug 24 2008: (Start)
O.g.f.: 2*(1 + 2*x + 5*x^2)/(1-x)^3.
a(n) = 2*A053755(n). (End)
a(n) = a(-n); a(n) + a(-n) = A158444(n). - Bruno Berselli, Sep 06 2011
a(n) = 2*(A000466(n) + 2). - Martin Ettl, Nov 12 2012
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: 2*exp(x)*(1 + 4*x + 4*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A001386 Coordination sequence for 4-dimensional I-centered tetragonal orthogonal lattice.

Original entry on oeis.org

1, 12, 56, 164, 368, 700, 1192, 1876, 2784, 3948, 5400, 7172, 9296, 11804, 14728, 18100, 21952, 26316, 31224, 36708, 42800, 49532, 56936, 65044, 73888, 83500, 93912, 105156, 117264, 130268, 144200, 159092, 174976, 191884, 209848, 228900, 249072, 270396, 292904

Offset: 0

N. J. A. Sloane

			G.f.: (1+x)^2*(1+6*x+x^2)/(1-x)^4. - _Colin Barker_, Apr 14 2012

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hamzeh Mujahed and Benedek Nagy, Wiener Index on Lines of Unit Cells of the Body-Centered Cubic Grid, Mathematical Morphology and Its Applications to Signal and Image Processing, 12th International Symposium, ISMM 2015.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Equals 4 * A004006(2n), n>0.

Maple
```
[ seq( (16*k^3+20*k)/3, k=1..40) ];
```

CoefficientList[Series[(1+x)^2*(1+6*x+x^2)/(1-x)^4,{x,0,30}],x] (* Vincenzo Librandi, Apr 15 2012 *)

A008355 Coordination sequence for D_5 lattice.

Original entry on oeis.org

1, 40, 370, 1640, 4930, 11752, 24050, 44200, 75010, 119720, 182002, 265960, 376130, 517480, 695410, 915752, 1184770, 1509160, 1896050, 2353000, 2888002, 3509480, 4226290, 5047720, 5983490, 7043752

Offset: 0

N. J. A. Sloane and J. H. Conway

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

A row of array A103903.

[1]cat[2*(9*n^2+1)*(n^2+1): n in [1..30]]; // Vincenzo Librandi, Apr 16 2012

Maple
```
2*(9*n^2+1)*(n^2+1);
```

CoefficientList[Series[(1+x)*(1+34*x+146*x^2+34*x^3+x^4)/(1-x)^5,{x,0,30}],x] (* Vincenzo Librandi, Apr 16 2012 *)

a(n) = 2*(9*n^2+1)*(n^2+1) (see MAPLE line).

G.f.: (1+x)*(1+34*x+146*x^2+34*x^3+x^4)/(1-x)^5. [Colin Barker, Apr 14 2012]

A008357 Coordination sequence for D_6 lattice.

Original entry on oeis.org

1, 60, 792, 4724, 18096, 52716, 127816, 271908, 524640, 938652, 1581432, 2537172, 3908624, 5818956, 8413608, 11862148, 16360128, 22130940, 29427672, 38534964, 49770864, 63488684, 80078856

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

A row of array A103903.

Maple
```
4/15*n*(37+130*n^2+58*n^4);
```

G.f.: (x^6+54*x^5+447*x^4+852*x^3+447*x^2+54*x+1)/(x-1)^6. [Colin Barker, Sep 26 2012]

cp's OEIS Frontend

A008383 Coordination sequence for A_4 lattice.

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A008385 Coordination sequence for A_5 lattice.

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A008531 Coordination sequence for {A_4}* lattice.

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A008349 Crystal ball sequence for E_8 lattice.

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A008387 Coordination sequence for A_6 lattice.

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A008528 Coordination sequence for 4-dimensional RR-centered di-isohexagonal orthogonal lattice.

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A108100 a(n) = (2n-1)^2 + (2n+1)^2.