A019558 -id:A019558 - cp's OEIS frontend

A100178 Structured hexagonal diamond numbers (vertex structure 5).

1, 8, 29, 72, 145, 256, 413, 624, 897, 1240, 1661, 2168, 2769, 3472, 4285, 5216, 6273, 7464, 8797, 10280, 11921, 13728, 15709, 17872, 20225, 22776, 25533, 28504, 31697, 35120, 38781, 42688, 46849, 51272, 55965, 60936, 66193, 71744, 77597, 83760, 90241, 97048, 104189

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Row 1 of the convolution array A213838. - Clark Kimberling, Jul 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578 (alternate vertex), A000447 (structured diamonds) A100145 (for more on structured numbers).

Cf. A019558, A167471, A213838.

[(1/6)*(8*n^3-6*n^2+4*n): n in [1..40]]; // Vincenzo Librandi, Aug 03 2011

LinearRecurrence[{4, -6, 4, -1}, {1, 8, 29, 72}, 50] (* Paolo Xausa, Aug 06 2025 *)

a(n) = (1/6)*(8*n^3 - 6*n^2 + 4*n).
G.f.: x*(1+4*x+3*x^2)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 04 2012
From Elmo R. Oliveira, Aug 28 2025: (Start)
E.g.f.: exp(x)*x*(4*x^2 + 9*x + 3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
a(n) = A167471(n)/16 = A019558(n)/48. (End)

A008340 Coordination sequence for E_8 lattice.

Original entry on oeis.org

1, 240, 9120, 121680, 864960, 4113840, 14905440, 44480400, 114879360, 265422960, 561403680, 1105317840, 2050966080, 3620750640, 6126497760, 9994133520, 15792541440, 24266930160, 36377039520, 53340513360, 76681767360

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, 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).
G. Nebe and N. J. A. Sloane, Home page for this lattice
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 (8,-28,56,-70,56,-28,8,-1).

Cf. A019557, A019558, A008397, A008399.

if n = 0 then 1 else 456/7*n^7-120*n^6+312*n^5-120*n^4-48*n^3+240*n^2-624/7*n;

Join[{1},Table[456/7*n^7-120*n^6+312*n^5-120*n^4-48*n^3+ 240*n^2- 624/7*n,{n,20}]] (* Harvey P. Dale, Jul 14 2014 *)

a(n) = if n = 0 then 1 else (456/7)*n^7-120*n^6+312*n^5-120*n^4-48*n^3+240*n^2-(624/7)*n.
Bacher et al. give a g.f.
G.f.: (x^8 +232*x^7 +24508*x^6 +107224*x^5 +133510*x^4 +55384*x^3 +7228*x^2 +232*x +1)/(x -1)^8 = 1 + 240*x* (1+30*x+231*x^2+556*x^3+447*x^4+102*x^5+x^6) /(1-x)^8. [Colin Barker, Sep 26 2012]

The values given by O'Keeffe are incorrect.

A008399 Coordination sequence for E_6 lattice.

Original entry on oeis.org

1, 72, 1062, 6696, 26316, 77688, 189810, 405720, 785304, 1408104, 2376126, 3816648, 5885028, 8767512, 12684042, 17891064, 24684336, 33401736, 44426070, 58187880, 75168252, 95901624, 120978594, 151048728, 186823368, 229078440, 278657262, 336473352, 403513236

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 (6,-15,20,-15,6,-1).

Cf. A008397, A008340, A019557, A019558.

[1] cat [9*n*(13*n^2+7)*(n^2+1)/5: n in [1..40]]; // G. C. Greubel, May 29 2023

1, seq(117/5*n^5+36*n^3+63/5*n, n=1..30);

LinearRecurrence[{6,-15,20,-15,6,-1},{1,72,1062,6696,26316,77688, 189810},30] (* Harvey P. Dale, Oct 24 2022 *)

[9*n*(13*n^2+7)*(n^2+1)//5 +int(n==0) for n in range(41)] # G. C. Greubel, May 29 2023

a(n) = 9*n*(13*n^2+7)*(n^2+1)/5 for n >= 1.
Bacher et al. give a g.f.
G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^6 = 1 + 18*x*(4+35*x+78*x^2+35*x^3+4*x^4)/(1-x)^6. - Colin Barker, Sep 26 2012
E.g.f.: 1 + (1/5)*x*(360 + 2295*x + 3105*x^2 + 1170*x^3 + 117*x^4 )*exp(x). - G. C. Greubel, May 29 2023

A167471 Janet periodic table of the elements and structured hexagonal diamond numbers. a(n) = A166911(2n) + A166911(2n+1).

Original entry on oeis.org

16, 128, 464, 1152, 2320, 4096, 6608, 9984, 14352, 19840, 26576, 34688, 44304, 55552, 68560, 83456, 100368, 119424, 140752, 164480, 190736, 219648, 251344, 285952, 323600, 364416, 408528, 456064, 507152, 561920, 620496, 683008, 749584, 820352, 895440, 974976

Offset: 1

Paul Curtz, Nov 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A019558, A100178, A166464, A166911.

[16*n*(2-3*n+4*n^2)/3: n in [1..40]]; // Vincenzo Librandi, Aug 03 2011

Table[16*n*(2 - 3*n + 4*n^2)/3, {n,1,50}] (* G. C. Greubel, Jun 13 2016 *)

a(n) = 16*n*(2 - 3*n + 4*n^2)/3 = 16*A100178(n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(1)=16, a(2)=128, a(3)=464, a(4)=1152.
From Elmo R. Oliveira, Aug 28 2025: (Start)
G.f.: 16*x*(1 + 4*x + 3*x^2)/(1 - x)^4.
E.g.f.: 16*x*(3 + 9*x + 4*x^2)*exp(x)/3.
a(n) = A019558(n)/3. (End)

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A100178 Structured hexagonal diamond numbers (vertex structure 5).

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A008340 Coordination sequence for E_8 lattice.

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A008399 Coordination sequence for E_6 lattice.

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A167471 Janet periodic table of the elements and structured hexagonal diamond numbers. a(n) = A166911(2*n) + A166911(2*n+1).

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A167471 Janet periodic table of the elements and structured hexagonal diamond numbers. a(n) = A166911(2n) + A166911(2n+1).