A005925 -id:A005925 - cp's OEIS frontend

A217512 Partial sums of nonzero terms in A005925.

1, 5, 17, 29, 35, 47, 71, 87, 99, 123, 147, 159, 167, 191, 239, 275, 281, 293, 329, 357, 381, 417, 441, 465, 489, 525, 597, 633, 657, 705, 729, 741, 801, 849, 885, 915, 943, 1015, 1063, 1087, 1099, 1147, 1207, 1231, 1291, 1339, 1363, 1371, 1419, 1503, 1551

Offset: 0

N. J. A. Sloane, Oct 05 2012

nonn

Sizes of clusters of atoms in diamond structure.

N. J. A. Sloane, Theta-Series and Magic Numbers for Diamond and Certain Ionic Crystal Structures, J. Math. Phys., 28 (1987), pp. 1653-1657.

Cf. A005925-A005927, A217512-A217514.

lista(m) = {nm = sqrtint(m)+1; T2 = sum(i=0, nm, 2*x^((i+1/2)^2 - 1/4)); T23 = T2^3; T3 = 1+ sum(i=1, nm, 2*x^(i^2)); T4 = subst(T3, x, -x); T33 = T3^3; T334 = sum(i=0, m, polcoeff(T33, i)*x^(4*i)); T43 = T4^3; T434 = sum(i=0, m, polcoeff(T43, i)*x^(4*i)); P = sum(i=0, poldegree(T23), polcoeff(T23,i)*x^(numerator(i+3/4))); D = (P + T334 + T434)/2; s = 0; for (i=0, m, v = polcoeff(D, i); if (v != 0, s += v; print1(s, ", ");););} \\ Michel Marcus, Feb 19 2013

More terms from Michel Marcus, Feb 19 2013

A077307 Partial sums of theta series of diamond (A005925).

Original entry on oeis.org

1, 1, 1, 5, 5, 5, 5, 5, 17, 17, 17, 29, 29, 29, 29, 29, 35, 35, 35, 47, 47, 47, 47, 47, 71, 71, 71, 87, 87, 87, 87, 87, 99, 99, 99, 123, 123, 123, 123, 123, 147, 147, 147, 159, 159, 159, 159, 159, 167, 167, 167, 191, 191, 191, 191, 191, 239, 239, 239, 275, 275, 275, 275, 275, 281, 281, 281, 293

Offset: 0

Jonathan Vos Post, Dec 13 2010

nonn

Cf. A005925.

a(n) = Sum_{i=0..n} A005925(i).

Incorrect a(16)=35 removed by Georg Fischer, Aug 29 2020

A217514 Partial sums of nonzero terms in A005927.

Original entry on oeis.org

4, 10, 22, 30, 42, 66, 82, 106, 136, 148, 172, 196, 220, 256, 268, 316, 344, 368, 404, 452, 476, 512, 542, 578, 610, 634, 706, 730, 790, 838, 874, 922, 950, 974, 1022, 1034, 1130, 1190, 1214, 1274, 1346, 1370, 1418, 1472, 1520, 1568, 1604, 1676, 1724, 1784

Offset: 0

N. J. A. Sloane, Oct 05 2012

nonn

Sizes of clusters of atoms in diamond structure.

N. J. A. Sloane, Theta-Series and Magic Numbers for Diamond and Certain Ionic Crystal Structures, J. Math. Phys., 28 (1987), pp. 1653-1657.

Cf. A005925-A005927, A217512-A217514.

lista(m) = {s = 0; for (n=0, m, v = A005927(n); if (v != 0, s += v; print1(s, ", ");););} \\ Michel Marcus, Feb 17 2013

More terms from Michel Marcus, Feb 17 2013

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

A217513 Partial sums of nonzero terms in A005926.

Original entry on oeis.org

2, 8, 20, 32, 38, 56, 74, 86, 116, 130, 136, 166, 190, 208, 238, 264, 288, 318, 342, 360, 384, 420, 444, 492, 522, 534, 588, 612, 648, 690, 714, 732, 768, 822, 840, 894, 942, 966, 1026, 1050, 1068, 1134, 1190, 1220, 1256, 1280, 1310, 1382, 1418, 1448, 1508

Offset: 0

N. J. A. Sloane, Oct 05 2012

nonn

Sizes of clusters of atoms in diamond structure.

Andy Huchala, Table of n, a(n) for n = 0..98
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A005925-A005927, A217512-A217514.

More terms from Andy Huchala, May 17 2023

A217511 Theta series of hexagonal diamond or Lonsdaleite net with respect to an atom.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 18

Offset: 0

N. J. A. Sloane, Oct 05 2012

nonn

Eq. (20) of [Sloane, 1987] gives the g.f. of this sequence if one replaces the binomial in the round brackets with the factor eta_{3/8}(X^(16/3)); this error propagated from Eq. (67) of [Sloane & Teo, 1985], where the second curly brackets should be replaced by psi_{8/3}(q^(16/3)) to get the g.f. of A005873 (or, alternatively, replace the power 4/3 with 1/3 in both formulas). - Andrey Zabolotskiy, Jun 04 2022

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
G. L. Hall, Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From _N. J. A. Sloane_, Dec 18 2012
N. J. A. Sloane, Theta-Series and Magic Numbers for Diamond and Certain Ionic Crystal Structures, J. Math. Phys., 28 (1987), pp. 1653-1657.
N. J. A. Sloane and Boon K. Teo, Theta series and magic numbers for closepacked spherical clusters, J. Chemical Phys 83, 6520-6534 (1985).

Cf. A005925, A008264, A004012, A005873.

a(n) = A004012(n/8) + A005873(n), where the 1st term is 0 unless 8|n. - Andrey Zabolotskiy, Jun 03 2022

Missing a(71) = 0 inserted by Andrey Zabolotskiy, Jun 03 2022

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}];

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A217512 Partial sums of nonzero terms in A005925.

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A077307 Partial sums of theta series of diamond (A005925).

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A217514 Partial sums of nonzero terms in A005927.

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

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A217513 Partial sums of nonzero terms in A005926.

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A217511 Theta series of hexagonal diamond or Lonsdaleite net with respect to an atom.

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A290705 Theta series of triamond.

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