Displaying 1-10 of 33 results found.
a(n) = 6*n^2 + 2 for n > 0, a(0)=1.
(Formerly M4497)
+10
580
1, 8, 26, 56, 98, 152, 218, 296, 386, 488, 602, 728, 866, 1016, 1178, 1352, 1538, 1736, 1946, 2168, 2402, 2648, 2906, 3176, 3458, 3752, 4058, 4376, 4706, 5048, 5402, 5768, 6146, 6536, 6938, 7352, 7778, 8216, 8666, 9128, 9602, 10088, 10586
COMMENTS
Number of points on surface of 3-dimensional cube in which each face has a square grid of dots drawn on it (with n+1 points along each edge, including the corners).
Coordination sequence for b.c.c. lattice.
Also coordination sequence for 3D uniform tiling with tile an equilateral triangular prism. - N. J. A. Sloane, Feb 06 2018 Binomial transform of [1, 7, 11, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Oct 22 2007 Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=2, s=1. After 8, all terms are in A000408. - Bruno Berselli, Feb 07 2012 For n > 0, the sequence of last digits (i.e., a(n) mod 10) is (8, 6, 6, 8, 2) repeating forever. - M. F. Hasler, Apr 05 2016 Number of cubes of edge length 1 required to make a hollow cube of edge length n+1. - Peter M. Chema, Apr 01 2017
REFERENCES
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (194) hP4
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #11.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(0) = 1, a(n) = (n+1)^3 - (n-1)^3. - Ilya Nikulshin (ilyanik(AT)gmail.com), Aug 11 2009
a(0)=1, a(1)=8, a(2)=26, a(3)=56; for n>3, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 25 2011 E.g.f.: 2*(1 + 3*x + 3*x^2)*exp(x) - 1. - G. C. Greubel, Dec 01 2017 Sum_{n>=0} 1/a(n) = 3/4+ Pi*sqrt(3)*coth(Pi/sqrt 3)/12 = 1.2282133.. - R. J. Mathar, Apr 27 2024
EXAMPLE
For n = 1 we get the 8 corners of the cube; for n = 2 each face has 9 points, for a total of 8 + 12 + 6 = 26.
MAPLE
A005897:=-(z+1)*(z**2+4*z+1)/(z-1)**3; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Join[{1}, 6Range[50]^2+2] (* or *) Join[{1}, LinearRecurrence[{3, -3, 1}, {8, 26, 56}, 50]] (* Harvey P. Dale, Oct 25 2011 *)
PROG
(PARI) x='x+O('x^30); Vec(serlaplace(2*(1 + 3*x + 3*x^2)*exp(x) - 1)) \\ G. C. Greubel, Dec 01 2017 (Haskell) a005897 n = if n == 0 then 1 else 6 * n ^ 2 + 2 -- Reinhard Zumkeller, Apr 27 2014
CROSSREFS
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).
(Formerly M3380)
+10
84
1, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234
COMMENTS
Number of n-matchings of the wheel graph W_{2n} (n > 0). Example: a(2)=10 because in the wheel W_4 (rectangle ABCD and spokes OA,OB,OC,OD) we have the 2-matchings: (AB, OC), (AB, OD), (BC, OA), (BC,OD), (CD,OA), (CD,OB), (DA,OB), (DA,OC), (AB,CD) and (BC,DA). - Emeric Deutsch, Dec 25 2004 For n > 0 a(n) is the difference of two tetrahedral (or pyramidal) numbers: binomial(n+3, 3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n+1) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-3)(n-2)(n-1)/6. - Alexander Adamchuk, May 20 2006; updated by Peter Munn, Aug 25 2017 due to changed offset in A000292Equals binomial transform of [1, 3, 3, 1, -1, 1, -1, 1, -1, 1, ...]. Binomial transform of A005893 = nonzero terms of A053545: (1, 5, 19, 63, 191, ...). - Gary W. Adamson, Apr 28 2008 Disregarding the terms < 10, the sums of four consecutive triangular numbers ( A000217). - Rick L. Shepherd, Sep 30 2009 Use a set of n concentric circles where n >= 0 to divide the plane. a(n) is the maximal number of regions after the 2nd division. - Frank M Jackson, Sep 07 2011 Euler transform of length 4 sequence [4, 0, 0, -1]. - Michael Somos, May 14 2014 Also, growth series for affine Coxeter group (or affine Weyl group) A_3 or D_3. - N. J. A. Sloane, Jan 11 2016 For n > 2 the generalized Pell's equation x^2 - 2*(a(n) - 2)y^2 = (a(n) - 4)^2 has a finite number of positive integer solutions. - Muniru A Asiru, Apr 19 2016
REFERENCES
N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #28.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf). Reticular Chemistry Structure Resource, sod.
FORMULA
G.f.: (1 - x^4)/(1-x)^4.
a(n) = binomial(n+3, 3) - binomial(n-1, 3) for n >= 1. - Mitch Harris, Jan 08 2008 a(n) = (n+1)^2 + (n-1)^2. - Benjamin Abramowitz, Apr 14 2009
a(0)=1, a(1)=4, a(2)=10, a(3)=20, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 26 2012 For n >= 2: a(n) = a(n-1) + 4*n - 2. - Bob Selcoe, Mar 22 2016 Sum_{n>=0} 1/a(n) = (coth(Pi)*Pi + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi)*Pi + 3)/4. (End)
Empirical: Integral_{u=-oo..+oo} sigmoid(u)*log(sigmoid(n * u)) du = -Pi^2*a(n) / (24*n), where sigmoid(x) = 1/(1+exp(-x)). Also works for non-integer n>0. - Carlo Wood, Dec 04 2023 Let P(k,n) be the n-th k-gonal number. Then P(a(k),n) = (k*n-k+1)^2 + (k-1)^2*(n-1). - Charlie Marion, May 15 2024
EXAMPLE
G.f. = 1 + 4*x + 10*x^2 + 20*x^3 + 34*x^4 + 52*x^5 + 74*x^6 + 100*x^7 + ...
MATHEMATICA
Join[{1}, LinearRecurrence[{3, -3, 1}, {4, 10, 20}, 50]] (* Harvey P. Dale, Feb 26 2012 *) a[ n_] := SeriesCoefficient[ (1 - x^4) / (1 - x)^4, {x, 0, Abs@n}]; (* Michael Somos, May 14 2014 *) a[ n_] := 2 n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 14 2014 *)
CROSSREFS
Cf. similar sequences listed in A255843. The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.
(Formerly M4115)
+10
75
1, 6, 18, 38, 66, 102, 146, 198, 258, 326, 402, 486, 578, 678, 786, 902, 1026, 1158, 1298, 1446, 1602, 1766, 1938, 2118, 2306, 2502, 2706, 2918, 3138, 3366, 3602, 3846, 4098, 4358, 4626, 4902, 5186, 5478, 5778, 6086, 6402, 6726, 7058, 7398, 7746, 8102, 8466
COMMENTS
Also, the number of regions the plane can be cut into by two overlapping concave (2n)-gons. - Joshua Zucker, Nov 05 2002 If X is an n-set and Y_i (i=1,2,3) are mutually disjoint 2-subsets of X then a(n-5) is equal to the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007 Also the least number of unit cubes required, at the n-th iteration, to surround a 3D solid built from unit cubes, in order to hide all its visible faces, starting with a unit cube. - R. J. Cano, Sep 29 2015 Also, coordination sequence for "tfs" 3D uniform tiling. - N. J. A. Sloane, Feb 10 2018 Also, the number of n-th order specular reflections arriving at a receiver point from an emitter point inside a cuboid with reflective faces. - Michael Schutte, Sep 18 2018
REFERENCES
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (225) cF8
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #16 and #22.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf).
FORMULA
Binomial transform of [1, 5, 7, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 02 2007 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=1, a(1)=6, a(2)=18, a(3)=38. - Harvey P. Dale, Nov 08 2011 Recurrence: n*a(n) = (n-2)*a(n-2) + 6*a(n-1), a(0)=1, a(1)=6. - Fung Lam, Apr 15 2014 a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=3, n>0. - Shel Kaphan, Feb 16 2023 Sum_{n>=0} 1/a(n) = 3/4 + Pi *sqrt(2)*coth( Pi/sqrt 2)/8 = 1.31858... - R. J. Mathar, Apr 27 2024
MATHEMATICA
Join[{1}, 4Range[40]^2+2] (* or *) Join[{1}, LinearRecurrence[{3, -3, 1}, {6, 18, 38}, 40]] (* Harvey P. Dale, Nov 08 2011 *)
PROG
(PARI) Vec(((1+x)/(1-x))^3 + O(x^100)) \\ Altug Alkan, Oct 26 2015
CROSSREFS
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Number of points on surface of cuboctahedron (or icosahedron): a(0) = 1; for n > 0, a(n) = 10n^2 + 2. Also coordination sequence for f.c.c. or A_3 or D_3 lattice.
(Formerly M4834)
+10
64
1, 12, 42, 92, 162, 252, 362, 492, 642, 812, 1002, 1212, 1442, 1692, 1962, 2252, 2562, 2892, 3242, 3612, 4002, 4412, 4842, 5292, 5762, 6252, 6762, 7292, 7842, 8412, 9002, 9612, 10242, 10892, 11562, 12252, 12962, 13692, 14442, 15212, 16002
COMMENTS
Sequence found by reading the segment (1, 12) together with the line from 12, in the direction 12, 42, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
REFERENCES
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (225) cF4
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #1.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
S. Rosen, Wizard of the Dome: R. Buckminster Fuller; Designer for the Future. Little, Brown, Boston, 1969, p. 109.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf).
FORMULA
G.f.: (1+x)*(1+8*x+x^2)/(1-x)^3. - Simon Plouffe in his 1992 dissertation G.f. for coordination sequence for A_n lattice is (1-z)^(-n) * Sum_{i=0..n} binomial(n, i)^2*z^i. [Bacher et al.]
a(n) = 12 + 24*(n-1) + 8* A000217(n-2) + 6* A000290(n-1). The properties of the cuboctahedron, namely, its number of vertices (12), edges (24), and faces as well as face-type (8 triangles and 6 squares), are involved in this formula. - Peter M. Chema, Mar 26 2017 E.g.f.: -1 + 2*(1 + 5*x + 5*x^2)*exp(x). - G. C. Greubel, May 25 2023 Sum{n>=0} 1/a(n) = 3/4 + Pi*sqrt(5)*coth(Pi/sqrt 5)/20 = 1.14624... - R. J. Mathar, Apr 27 2024
MATHEMATICA
Join[{1}, 10*Range[40]^2+2] (* or *) Join[{1}, LinearRecurrence[{3, -3, 1}, {12, 42, 92}, 40]] (* Harvey P. Dale, May 28 2014 *)
PROG
(PARI) a(n)=if(n<0, 0, 10*n^2+1+(n>0))
(Magma) [n eq 0 select 1 else 2*(5*n^2+1): n in [0..55]]; // G. C. Greubel, May 25 2023 (SageMath) [2*(5*n^2 + 1)-int(n==0) for n in range(56)] # G. C. Greubel, May 25 2023
CROSSREFS
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Coordination sequence for hexagonal close-packing.
+10
52
1, 12, 44, 96, 170, 264, 380, 516, 674, 852, 1052, 1272, 1514, 1776, 2060, 2364, 2690, 3036, 3404, 3792, 4202, 4632, 5084, 5556, 6050, 6564, 7100, 7656, 8234, 8832, 9452, 10092, 10754, 11436, 12140, 12864, 13610, 14376, 15164, 15972, 16802, 17652, 18524, 19416, 20330, 21264, 22220
REFERENCES
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #2.
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf).
FORMULA
a(n) = floor( 21*n^2 / 2 ) + 2, for n>= 1.
G.f.: (x^4 +10*x^3 +20*x^2 +10*x +1)/((1+x)*(1-x)^3).
a(0)=1, a(1)=12, a(2)=44, a(3)=96, a(4)=170, a(n)=2*a(n-1)-2*a(n-3)+ a(n-4). - Harvey P. Dale, Feb 15 2014 E.g.f.: ((4 + 21*x + 21*x^2)*cosh(x) + 3*(1 + 7*x + 7*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Mar 14 2024
MATHEMATICA
Join[{1}, Floor[(21Range[40]^2)/2]+2] (* or *) Join[{1}, LinearRecurrence[ {2, 0, -2, 1}, {12, 44, 96, 170}, 40]] (* Harvey P. Dale, Feb 15 2014 *) CoefficientList[Series[(x^4 + 10 x^3 + 20 x^2 + 10 x + 1)/(1 - x)^3/(x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 16 2014 *)
PROG
(Magma) I:=[1, 12, 44, 96, 170]; [n le 5 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 16 2014 (PARI) for(n=0, 50, print1(if(n==0, 1, 2 + floor(21*n^2/2)), ", ")) \\ G. C. Greubel, Feb 20 2018 (Magma) [1] cat [2 + Floor(21*n^2/2): n in [1..50]]; // G. C. Greubel, Feb 20 2018
CROSSREFS
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
a(0) = 1; for n>0, a(n) = 41*n^2 + 2.
+10
35
1, 43, 166, 371, 658, 1027, 1478, 2011, 2626, 3323, 4102, 4963, 5906, 6931, 8038, 9227, 10498, 11851, 13286, 14803, 16402, 18083, 19846, 21691, 23618, 25627, 27718, 29891, 32146, 34483, 36902, 39403, 41986, 44651, 47398, 50227, 53138, 56131, 59206, 62363
COMMENTS
Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=3, s=4. After 1, all terms are in A000408.
FORMULA
O.g.f.: (1 + x)*(1 + 39*x + x^2)/(1 - x)^3.
MATHEMATICA
Join[{1}, 41 Range[39]^2 + 2]
CoefficientList[Series[(1 + x) (1 + 39 x + x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)
PROG
(Magma) [n eq 0 select 1 else 41*n^2+2: n in [0..39]];
(Magma) I:=[1, 43, 166, 371]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..41]]; // Vincenzo Librandi, Aug 18 2013
CROSSREFS
Sequences of the same type: A005893, A005897, A005899, A005901, A005903, A005905, A005914, A005918, A005919, A008527, A010000- A010023.
Coordination sequence for 4-dimensional cubic lattice (points on surface of 4-dimensional cross-polytope).
+10
20
1, 8, 32, 88, 192, 360, 608, 952, 1408, 1992, 2720, 3608, 4672, 5928, 7392, 9080, 11008, 13192, 15648, 18392, 21440, 24808, 28512, 32568, 36992, 41800, 47008, 52632, 58688, 65192, 72160, 79608, 87552, 96008, 104992, 114520, 124608, 135272
COMMENTS
Coordination sequence for 4-dimensional cyclotomic lattice Z[zeta_8].
If Y_i (i=1,2,3,4) are 2-blocks of a (n+4)-set X then a(n-3) is the number of 7-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Oct 28 2007
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf).
FORMULA
G.f.: ((1+x)/(1-x))^4.
a(n) = 8*n*(n^2+2)/3 for n>1.
a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=4, for n>=1. - Shel Kaphan, Feb 14 2023 E.g.f.: 1 + 8*exp(x)*x*(3 + 3*x + x^2)/3. - Stefano Spezia, Mar 14 2024
MATHEMATICA
CoefficientList[Series[((1+x)/(1-x))^4, {x, 0, 40}], x] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 8, 32, 88, 192}, 41] (* Harvey P. Dale, Jun 10 2011 *) f[n_] := 8 n (n^2 + 2)/3; f[0] = 1; Array[f, 38, 0] (* or *)
g[n_] := 4n^2 +2; f[n_] := f[n-1] + g[n] + g[n -1]; f[0] = 1; f[1] = 8; Array[f, 38, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PROG
(Magma) I:=[1, 8, 32, 88, 192]; [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..50]]; // Vincenzo Librandi, Jan 15 2018
a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.
+10
5
1, 34, 130, 290, 514, 802, 1154, 1570, 2050, 2594, 3202, 3874, 4610, 5410, 6274, 7202, 8194, 9250, 10370, 11554, 12802, 14114, 15490, 16930, 18434, 20002, 21634, 23330, 25090, 26914, 28802, 30754, 32770, 34850, 36994, 39202, 41474, 43810, 46210, 48674, 51202
COMMENTS
Sequence found by reading the line segment from 1 to 34 together with the line from 34, in the direction 34, 130, ..., in the rectangular spiral whose vertices are the generalized 18-gonal numbers A274979. The spiral begins as follows:
46_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _18
| |
| 0 |
| |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
| 1 15
|
51
(End)
FORMULA
Sum_{n>=0} 1/a(n) = 3/4 + Pi/16*coth(Pi/4) = 1.04940725316131.. - R. J. Mathar, May 07 2024
MATHEMATICA
CoefficientList[Series[(1 + x) (1 + 30 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2014 *)
CROSSREFS
Cf. A274979 (generalized 18-gonal numbers).
Coordination sequence for D_4 lattice.
+10
4
1, 24, 144, 456, 1056, 2040, 3504, 5544, 8256, 11736, 16080, 21384, 27744, 35256, 44016, 54120, 65664, 78744, 93456, 109896, 128160, 148344, 170544, 194856, 221376, 250200, 281424, 315144, 351456
REFERENCES
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf).
FORMULA
G.f.: (1+54*x^2+20*x+20*x^3+x^4)/(1-x)^4 = 1+24*x*(x+1)^2/(x-1)^4.
G.f. for coordination sequence of D_n lattice: (Sum(binomial(2*n, 2*i)*z^i, i=0..n)-2*n*z*(1+z)^(n-2))/(1-z)^n.
MAPLE
if n=0 then 1 else 8*n*(2*n^2+1); fi;
Coordination sequence for E_8 lattice.
+10
4
1, 240, 9120, 121680, 864960, 4113840, 14905440, 44480400, 114879360, 265422960, 561403680, 1105317840, 2050966080, 3620750640, 6126497760, 9994133520, 15792541440, 24266930160, 36377039520, 53340513360, 76681767360
REFERENCES
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf).
FORMULA
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]
MAPLE
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;
MATHEMATICA
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 *)
EXTENSIONS
The values given by O'Keeffe are incorrect.
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