A222408 -id:A222408 - cp's OEIS frontend

A008531 Coordination sequence for {A_4}* lattice.

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

A008487 Expansion of (1-x^5) / (1-x)^5.

Original entry on oeis.org

1, 5, 15, 35, 70, 125, 205, 315, 460, 645, 875, 1155, 1490, 1885, 2345, 2875, 3480, 4165, 4935, 5795, 6750, 7805, 8965, 10235, 11620, 13125, 14755, 16515, 18410, 20445, 22625, 24955, 27440, 30085, 32895, 35875, 39030, 42365, 45885, 49595, 53500, 57605, 61915

Offset: 0

N. J. A. Sloane

Related to the 4-dimensional cyclotomic lattice Z[zeta_5] (or A_4^{*}).

Growth series of the affine Weyl group of type A4. - Paul E. Gunnells, Jan 06 2017

R. Bott, The geometry and the representation theory of compact Lie groups, in: Representation Theory of Lie Groups, in: London Math. Soc. Lecture Note Ser., vol. 34, Cambridge University Press, Cambridge, 1979, pp. 65-90.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.

Colin Barker, 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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A008498, A008531, A222408.

concatenation([1], List([1..50], n-> 5*n*(n^2 +5)/6)); # G. C. Greubel, Nov 07 2019

[1] cat [5*n*(n^2 +5)/6: n in [1..50]]; // G. C. Greubel, Nov 07 2019

1, seq(5*n*(n^2 +5)/6, n=1..50); # G. C. Greubel, Nov 07 2019

CoefficientList[Series[(1-x^5)/(1-x)^5, {x, 0, 50}], x] (* Stefano Spezia, Dec 30 2018 *)

Vec((1-x^5) / (1-x)^5+O(x^50)) \\ Charles R Greathouse IV, Sep 26 2012; corrected by Colin Barker, Jan 06 2017

[1]+[5*n*(n^2 +5)/6 for n in (1..50)] # G. C. Greubel, Nov 07 2019

a(n) is the sum of 5 consecutive tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6 = A000292(n) for n>0, a(0) = 1. a(n) = A000292(n-4) + A000292(n-3) + A000292(n-2) + A000292(n-1) + A000292(n) for n>0, a(0) = 1. - Alexander Adamchuk, May 20 2006
Equals binomial transform of [1, 4, 6, 4, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 29 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. - Colin Barker, Jan 06 2017
For n >= 1, a(n) = (5*n^3 + 25*n)/6. - Christopher Hohl, Dec 30 2018
E.g.f.: 1 + x*(30 + 15*x + 5*x^2)*exp(x)/6. - G. C. Greubel, Nov 07 2019

cp's OEIS Frontend

A008531 Coordination sequence for {A_4}* lattice.

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