A038620 -id:A038620 - cp's OEIS frontend

A344071 Number of 2n-step self-avoiding cycles on the Laves graph.

1, 1, 0, 0, 0, 10, 0, 14, 112, 282, 600, 2618, 9400, 27378, 93828, 332070, 1127840, 3818438, 13369560, 46776898, 163073440, 572984748, 2026468708, 7176533646

Offset: 0

Sean A. Irvine, May 09 2021

Sean A. Irvine, Java program (github)
Wikipedia, Laves graph

Cf. A046944, A344040, A344070, A001337, A038620.

Name clarified by Andrey Zabolotskiy, Jun 03 2021

A344040 Number of n-step self-avoiding walks on the Laves graph with no non-contiguous adjacencies.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 96, 192, 384, 738, 1446, 2832, 5544, 10818, 21138, 40980, 79848, 155406, 302526, 587232, 1141620, 2214102, 4300410, 8340876, 16188108, 31368204, 60840540, 117858042, 228471396, 442435860, 857273136, 1659450072, 3214094028, 6220079160

Offset: 0

Sean A. Irvine, May 07 2021

Sean A. Irvine, Java program (github)
Wikipedia, Laves graph

Cf. A046944, A336906, A336907, A344039, A344071, A038620.

Name clarified by Andrey Zabolotskiy, Jun 03 2021

A046944 Number of self-avoiding walks of length n on the Laves graph.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1506, 2982, 5904, 11688, 23094, 45678, 90000, 177660, 349938, 690192, 1359288, 2678808, 5271558, 10381926, 20419224, 40191084, 79025262, 155469228, 305582724, 600935844, 1180783482, 2321203446, 4559743116, 8960747616

Offset: 0

N. J. A. Sloane

The Leu reference gives 31 terms, but apparently is incorrect for a(25) onward.

Previous name: Number of self-avoiding walks of length n on hydrogen peroxide lattice.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..40 from Kuz'min et al. (terms up to 37 from Sean A. Irvine)
Sean A. Irvine, Java program (github).
M. D. Kuz'min, R. O. Kuzian and J. Richter, Ferromagnetism of the semi-simple cubic lattice, Eur. Phys. J. Plus 135, 750 (2020). See Table 4 (multiply by 2 to get this sequence).
J. A. Leu, Self-avoiding walks on a pair of three dimensional lattices, Phys. Lett., 29A (1969), 641-642.
J. A. Leu, D. D. Betts, and C. J. Elliott, High-temperature critical properties of the Ising model on a triple of related lattices, Canadian Journal of Physics, 47 (1969), 1671-1689.
Wikipedia, Laves graph

Cf. A344040, A344071, A038620, A046945.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
a(25) onward corrected by Sean A. Irvine, May 06 2021
Name clarified by Andrey Zabolotskiy, Jun 03 2021

A038621 Growth function of an infinite cubic graph (number of nodes at distance <=n from fixed node).

Original entry on oeis.org

1, 4, 10, 22, 46, 81, 129, 198, 284, 392, 530, 691, 883, 1114, 1374, 1674, 2022, 2405, 2837, 3326, 3856, 4444, 5098, 5799, 6567, 7410, 8306, 9278, 10334, 11449, 12649, 13942, 15300, 16752, 18306, 19931, 21659, 23498, 25414, 27442, 29590, 31821, 34173, 36654

Offset: 0

Jan Kristian Haugland

Partial sums of A038620.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A038620, A290705.

CoefficientList[Series[(x + 1) (2 x^8 - 4 x^7 + 3 x^6 - x^5 + 6 x^4 + 2 x^3 + 2 x^2 + x + 1)/((x - 1)^4 (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 22 2013 *)
LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{1,4,10,22,46,81,129,198,284,392},50] (* Harvey P. Dale, Sep 03 2016 *)

a(0)=1, a(1)=4; for n>=2: if n == 0 (mod 3), a(n) = (4*n^3 + 6*n^2 + 15*n - 9)/9; if n == 1 (mod 3), a(n) = (4*n^3 + 6*n^2 + 18*n - 10)/9; if n == 2 (mod 3), a(n) = (4*n^3 + 6*n^2 + 15*n + 4)/9.

G.f.: (x+1)*(2*x^8-4*x^7+3*x^6-x^5+6*x^4+2*x^3+2*x^2+x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, May 10 2013

More terms from Colin Barker, May 10 2013

A290705 Theta series of triamond.

Original entry on oeis.org

1, 3, 0, 6, 0, 6, 8, 12, 6, 9, 0, 6, 0, 18, 0, 12, 12, 12, 0, 18, 0, 12, 24, 12, 8, 21, 0, 24, 0, 6, 0, 24, 6, 24, 0, 12, 0, 30, 24, 12, 24, 12, 0, 30, 0, 30, 0, 24, 24, 27, 0, 12, 0, 18, 32, 36, 0, 24, 0, 18, 0, 30, 0, 36, 12, 12, 0, 42, 0, 24, 48, 12, 30

Offset: 0

Andrey Zabolotskiy, Aug 09 2017

nonn

Theta series with respect to a node of a lattice known as triamond, Laves graph [embedded in space], K_4 lattice, (10,3)-a, or srs net. This lattice possesses the "strong isotropic" property; the only other lattice that has this property in 3 dimensions is the diamond lattice. Unlike diamond, triamond is chiral.

A004013 and 3*A045828, interleaved.

Toshikazu Sunada, Crystals that nature might miss creating, Notices Amer. Math. Soc. 55 (No. 2, 2008), 208-215.
Toshikazu Sunada, Correction to "Crystals That Nature Might Miss Creating", Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343. [Annotated scanned copy]
Wikipedia, Laves graph

Cf. A004013, A005925, A045828, A113062.

See A038620 for coordination sequence.

(* count lattice sites straightforwardly *)
cell = Join @@ ({#, # + {1, 1, 1}/2} & /@ {{0, 0, 0}, {1/4, 0, 1/4}, {-1/4, -1/4, 0}, {0, 1/4, -1/4}}); (* lattice sites in a conventional bcc unit cell *)
n = 10;
s = O[q]^(n^2 + 1) + Sum[q^(8 Norm[a + {i, j, k}]^2), {i, -n-1, n+1}, {j, -n-1,  n+1}, {k, -n-1, n+1}, {a, cell}];
CoefficientList[Normal[s], q] &
(* or use the generation function *)
a[n_] := SeriesCoefficient[ EllipticTheta[3, 0, x^8]^3 + EllipticTheta[ 2, 0, x^8]^3 + 3/4 EllipticTheta[3, 0, x^2] EllipticTheta[2, 0, x^2]^2, {x, 0, n}];

A344126 Coordination sequence for the hypertriangular lattice.

Original entry on oeis.org

1, 6, 24, 48, 86, 138, 192, 260, 348, 432, 530, 654, 768, 896, 1056, 1200, 1358, 1554, 1728, 1916, 2148, 2352, 2570, 2838, 3072, 3320, 3624, 3888, 4166, 4506, 4800, 5108, 5484, 5808, 6146, 6558, 6912, 7280, 7728, 8112, 8510, 8994, 9408, 9836, 10356, 10800

Offset: 0

Sean A. Irvine, May 09 2021

Reticular Chemistry Structure Resource (RCSR), The lcy net
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A046945, A344039, A344070, A005901, A038620.

G.f.: (1+5*x+18*x^2+22*x^3+28*x^4+16*x^5+7*x^6-3*x^7+2*x^8) / ((x^2+x+1)^2 * (1-x)^3).

A126591 Coordination sequence of a point in the sphere packing 4/3/c27 of Fischer, or the lcv-a net.

Original entry on oeis.org

1, 4, 6, 11, 12, 22, 24, 44, 48, 88, 91, 128, 111, 182, 167, 254, 223, 296, 259, 392, 351, 516, 420, 536, 470, 682, 598, 850, 682, 872, 746, 1044, 908, 1264, 1007, 1280, 1087, 1502, 1283, 1750, 1395

Offset: 0

Wilfrid E. Klee (wiwi.klee(AT)inka.de), Mar 23 2007, Apr 03 2007

Space group of the packing is I4(1)32. The terms are computed using the TOPOLAN program.

W. Fischer, Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit drei Freiheitsgraden, Zeitschr. f. Kristallographie Vol. 140 (1974), pp. 50-74 (see p. 63).
R. W. Grosse-Kunstleve, G.O. Brunner and N. J. A. Sloane, Algebraic description of coordination sequences and exact topological densities for zeolites, Acta Cryst. Vol A52 (1996), pp. 879-889.
Reticular Chemistry Structure Resource, lcv-a.

Cf. A038620.

Name edited by Andrey Zabolotskiy, Oct 05 2017

cp's OEIS Frontend

A344071 Number of 2n-step self-avoiding cycles on the Laves graph.

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A344040 Number of n-step self-avoiding walks on the Laves graph with no non-contiguous adjacencies.

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A046944 Number of self-avoiding walks of length n on the Laves graph.

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A038621 Growth function of an infinite cubic graph (number of nodes at distance <=n from fixed node).

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A290705 Theta series of triamond.

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A344126 Coordination sequence for the hypertriangular lattice.

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A126591 Coordination sequence of a point in the sphere packing 4/3/c27 of Fischer, or the lcv-a net.

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Author

Keywords

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