A343599 -id:A343599 - cp's OEIS frontend

A005899 Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.

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

Offset: 0

N. J. A. Sloane

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
Binomial transform of a(n) is A055580(n). - Wesley Ivan Hurt, Apr 15 2014
The identity (4*n^2+2)^2 - (n^2+1)*(4*n)^2 = 4 can be written as a(n)^2 - A002522(n)*A008586(n)^2 = 4. - Vincenzo Librandi, Jun 15 2014
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

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).

T. D. Noe, Table of n, a(n) for n = 0..1000
Barry Balof, Restricted tilings and bijections, J. Integer Seq. 15 (2012), no. 2, Article 12.2.3, 17 pp.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences
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., A52 (1996), pp. 879-889.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
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]
Carlos I. Perez-Sanchez, The Spectral Action on quivers, arXiv:2401.03705 [math.RT], 2024.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Reticular Chemistry Structure Resource (RCSR), The pcu tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The tfs tiling (or net)
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
N. J. A. Sloane, Illustration of a(0)=1, a(1)=6, a(2)=18 (from Teo-Sloane 1985)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Partial sums give A001845.
Column 2 * 2 of array A188645.
Cf. A001844, A002522, A008586, A010014, A055580, A195322, A206399.
Cf. A000384, A014105, A000217, A008574, A008412, A035597.
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.
Row 3 of A035607, A266213, A343599.
Column 3 of A113413, A119800, A122542.

[4*n^2 + 2 : n in [0..50]]; // Wesley Ivan Hurt, Oct 26 2015

A005899:=n->4*n^2 + 2; seq(A005899(n), n=0..50); # Wesley Ivan Hurt, Apr 15 2014

Join[{1},4Range[40]^2+2] (* or *) Join[{1},LinearRecurrence[{3,-3,1},{6,18,38},40]] (* Harvey P. Dale, Nov 08 2011 *)

Vec(((1+x)/(1-x))^3 + O(x^100)) \\ Altug Alkan, Oct 26 2015

G.f.: ((1+x)/(1-x))^3. - Simon Plouffe in his 1992 dissertation
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
For n > 0, a(n) = A001844(n-1) + A001844(n) = (n-1)^2 + 2n^2 + (n+1)^2. - Doug Bell, Aug 18 2015
For n > 0, a(n) = A010014(n) - A195322(n). - R. J. Cano, Sep 29 2015
For n > 0, a(n) = A000384(n+1) + A014105(n-1). - Bruce J. Nicholson, Oct 08 2017
a(n) = A008574(n) + A008574(n-1) + a(n-1). - Bruce J. Nicholson, Dec 18 2017
a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=3, n>0. - Shel Kaphan, Feb 16 2023
a(n) = A035597(n)*3/n, for n>0. - Shel Kaphan, Feb 26 2023
E.g.f.: exp(x)*(2 + 4*x + 4*x^2) - 1. - Stefano Spezia, Mar 08 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

A035597 Number of points of L1 norm 3 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 12, 38, 88, 170, 292, 462, 688, 978, 1340, 1782, 2312, 2938, 3668, 4510, 5472, 6562, 7788, 9158, 10680, 12362, 14212, 16238, 18448, 20850, 23452, 26262, 29288, 32538, 36020, 39742, 43712, 47938, 52428, 57190, 62232, 67562

Offset: 0

N. J. A. Sloane

Sums of the first n terms > 0 of A001105 in palindromic arrangement. a(n) = Sum_{i=1 .. n} A001105(i) + Sum_{i=1 .. n-1} A001105(i), e.g. a(3) = 38 = 2 + 8 + 18 + 8 + 2; a(4) = 88 = 2 + 8 + 18 + 32 + 18 + 8 + 2. - Klaus Purath, Jun 19 2020

Apart from multiples of 3, all divisors of n are also divisors of a(n), i.e. if n is not divisible by 3, a(n) is divisible by n. All divisors d of a(n) for d !== 0 (mod) 3 are also divisors of a(abs(n-d)) and a(n+d). For all n congruent to 0,2,7 (mod 9) a(n) is divisible by 3. If n is divisible by 3^k, a(n) is divisible by 3^(k-1). - Klaus Purath, Jul 24 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII. Coordination sequences, Proc. Roy. Soc. Lond. A 458 (1996) 2369-2389.
Milzn Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
R. J. Mathar, Bivariate generating functions for non-attacking Wazirs on rectangular boards, viXra:2404.0122 (2024) page 14
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A069894.
Cf. A001105, A005900, A084570, A097869.
Cf. A001844, A001845, A005899.
Column 3 of A035607, A266213, A343599.
Row 3 of A113413, A119800, A122542.

[(4*n^3 + 2*n)/3: n in [0..40]]; // Vincenzo Librandi, Sep 19 2011

f := proc(n,m) local i; sum( 2^i*binomial(n,i)*binomial(m-1,i-1),i=1..min(n,m)); end; # n=dimension, m=norm

Table[(4n^3+2n)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,12,38},41] (* Harvey P. Dale, Sep 18 2011 *)

a(n) = (4*n^3 + 2*n)/3.
a(n) = 2*A005900(n). - R. J. Mathar, Dec 05 2009
a(0)=0, a(1)=2, a(2)=12, a(3)=38, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: (2*x*(x+1)^2)/(x-1)^4. - Harvey P. Dale, Sep 18 2011
a(n) = -a(-n), a(n+1) = A097869(4n+3) = A084570(2n+1). - Bruno Berselli, Sep 20 2011
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=3. Also, a(n) = A001845(n) - A001844(n). - Shel Kaphan, Feb 26 2023
a(n) = A005899(n)*n/3. - Shel Kaphan, Feb 26 2023
a(n) = A006331(n)+A006331(n-1). - R. J. Mathar, Aug 12 2025

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 16, 66, 192, 450, 912, 1666, 2816, 4482, 6800, 9922, 14016, 19266, 25872, 34050, 44032, 56066, 70416, 87362, 107200, 130242, 156816, 187266, 221952, 261250, 305552, 355266, 410816, 472642, 541200, 616962, 700416, 792066

Offset: 0

N. J. A. Sloane

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).
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A035596, A035597, A035599, A035600, A035601, A035602, A035603, A035604, A035605, A035606.
Cf. A001845, A001846, A008412, A014820.
Column 4 of A035607, A266213, A343599.
Row 4 of A113413, A119800, A122542.

[( 2*n^4 +4*n^2 )/3: n in [0..40]]; // Vincenzo Librandi, Apr 22 2012

f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm

CoefficientList[Series[2*x*(1+x)^3/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,16,66,192},50] (* Harvey P. Dale, Dec 11 2019 *)

a(n)=2*n^2*(n^2+2)/3 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = 2*n^2*(n^2 + 2)/3. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^3/(1-x)^5. - Colin Barker, Apr 15 2012
a(n) = 2*A014820(n-1). - R. J. Mathar, Dec 10 2013
a(n) = a(n-1) + A035597(n) + A035597(n-1). - Bruce J. Nicholson, Mar 11 2018
From Shel Kaphan, Feb 28 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=4.
a(n) = A001846(n) - A001845(n).
a(n) = A008412(n)*n/4. (End)
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 - 3*Pi*coth(sqrt(2)*Pi)/(8*sqrt(2)) + 3/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/16 + 3*Pi*cosech(sqrt(2)*Pi)/(8*sqrt(2)) - 3/16. (End)
E.g.f.: 2*exp(x)*x*(3 + 9*x + 6*x^2 + x^3)/3. - Stefano Spezia, Mar 14 2024

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A005899 Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.

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A035597 Number of points of L1 norm 3 in cubic lattice Z^n.

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A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

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