A206399 -id:A206399 - cp's OEIS frontend

A005897 a(n) = 6*n^2 + 2 for n > 0, a(0)=1.

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

Offset: 0

N. J. A. Sloane, Ralf W. Grosse-Kunstleve

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
First differences of A005898. - Jonathan Vos Post, Feb 06 2011
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
a(n) is the number of pieces on the outside of a (n+1) X (n+1) X (n+1) Rubik's cube. For n > 0: corners = 8, edges = 12*(n-1), center pieces = 6*(n-1)^2. - Demilade Runsewe, Jan 08 2025

			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.

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
John Elias, Illustration: Generalized octagonal cubes
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.
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]
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 hex tiling (or net)
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Index entries for sequences related to b.c.c. lattice
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000578, A206399.
See A005898 for partial sums.
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.

a005897 n = if n == 0 then 1 else 6 * n ^ 2 + 2 -- Reinhard Zumkeller, Apr 27 2014

[1] cat [6*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Oct 26 2011

A005897:=-(z+1)*(z**2+4*z+1)/(z-1)**3; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Join[{1},6Range[50]^2+2] (* or *) Join[{1},LinearRecurrence[{3,-3,1},{8,26,56},50]] (* Harvey P. Dale, Oct 25 2011 *)

a(n)=if(n,6*n^2+2,1) \\ Charles R Greathouse IV, Mar 06 2014

x='x+O('x^30); Vec(serlaplace(2*(1 + 3*x + 3*x^2)*exp(x) - 1)) \\ G. C. Greubel, Dec 01 2017

G.f.: (1+x)*(1+4*x+x^2)/(1-x)^3. - Simon Plouffe
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
a(n) = A033581(n) + 2. - Reinhard Zumkeller, Apr 27 2014
E.g.f.: 2*(1 + 3*x + 3*x^2)*exp(x) - 1. - G. C. Greubel, Dec 01 2017
a(n) = A000567(n+1) + A045944(n-1), for n>0. See illustration. - John Elias, Mar 12 2022
a(n) = 2*A056107(n), n>0. - R. J. Mathar, May 30 2022
Sum_{n>=0} 1/a(n) = 3/4+ Pi*sqrt(3)*coth(Pi/sqrt 3)/12 = 1.2282133.. - R. J. Mathar, Apr 27 2024
a(n) = 8 + 12*(n-1) + 6*(n-1)^2 for n > 0. - Demilade Runsewe, Jan 08 2025

A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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 A000292
Equals 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
Union of A188896, A277449, {1,4}. - Muniru A Asiru, Nov 25 2016
Interleaving of A008527 and A108099. - Bruce J. Nicholson, Oct 14 2019

			G.f. = 1 + 4*x + 10*x^2 + 20*x^3 + 34*x^4 + 52*x^5 + 74*x^6 + 100*x^7 + ...

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).

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).
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
J. M. Grau, C. Miguel, and A. M. Oller-Marcén, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017. See Theorem 1, p. 10.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy.]
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article 10.7.8.
M. O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst., A 47 (1991), 749-753.
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, sod.
Aditya Sivakumar and Dmitri Tymoczko, Intuitive Musical Homotopy, 2018.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558. DOI: 10.1021/ic00220a025.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000292, A053545, A206399.
Cf. similar sequences listed in A255843.
The growth series for the affine Coxeter groups D_3 through D_12 are A005893 and A266759-A266767.
For partial sums see A005894.
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.

[2*n^2-0^n+2: n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

A005893:=-(z+1)*(1+z^2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation

Join[{1}, Table[2*(n + 1)^2 + 2, {n, 0, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
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 *)

a(n)=2*n^2-0^n+2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1 - x^4)/(1-x)^4.
a(n) = A071619(n-1) + A071619(n) + A071619(n+1), n > 0. - Ralf Stephan, Apr 26 2003
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(n) = A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) for n >= 2. - Rick L. Shepherd, Sep 30 2009
a(n) = 2*n^2 - 0^n + 2. - Vincenzo Librandi, Sep 27 2011
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
a(n) = A228643(n+1,2) for n > 0. - Reinhard Zumkeller, Aug 29 2013
a(n) = a(-n) for all n in Z. - Michael Somos, May 14 2014
For n >= 2: a(n) = a(n-1) + 4*n - 2. - Bob Selcoe, Mar 22 2016
E.g.f.: -1 + 2*(1 + x + x^2)*exp(x). - Ilya Gutkovskiy, Apr 19 2016
a(n) = 2*A002522(n), n>0. - R. J. Mathar, May 30 2022
From Amiram Eldar, Sep 16 2022: (Start)
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

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.

Original entry on oeis.org

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

A005901 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.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, R. Vaughan

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

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).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
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.
G. Nebe and N. J. A. Sloane, Home page for this lattice
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]
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 fcu tiling (or net)
N. J. A. Sloane, A portion of the f.c.c. lattice packing.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
K. Urner, Microarchitecture of the Virus
R. Vaughan & N. J. A. Sloane, Correspondence, 1975
Wikipedia, Cuboctahedron
Index entries for sequences related to f.c.c. lattice
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A004015, A027599.
Cf. A033583, A062786, A092277, A206399.
Partial sums give A005902.
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.

[n eq 0 select 1 else 2*(5*n^2+1): n in [0..55]]; // G. C. Greubel, May 25 2023

Join[{1},10*Range[40]^2+2] (* or *) Join[{1},LinearRecurrence[{3,-3,1},{12,42,92},40]] (* Harvey P. Dale, May 28 2014 *)

PARI
```
a(n)=if(n<0,0,10*n^2+1+(n>0))
```

[2*(5*n^2 + 1)-int(n==0) for n in range(56)] # G. C. Greubel, May 25 2023

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+1) = A027599(n+2) + A092277(n+1) - Creighton Dement, Feb 11 2005
a(n) = 2 + A033583(n), n >= 1. - Omar E. Pol, Jul 18 2012
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
a(n) = A062786(n) + A062786(n+1). - R. J. Mathar, Feb 28 2018
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

A010001 a(0) = 1, a(n) = 5*n^2 + 2 for n>0.

Original entry on oeis.org

1, 7, 22, 47, 82, 127, 182, 247, 322, 407, 502, 607, 722, 847, 982, 1127, 1282, 1447, 1622, 1807, 2002, 2207, 2422, 2647, 2882, 3127, 3382, 3647, 3922, 4207, 4502, 4807, 5122, 5447, 5782, 6127, 6482, 6847, 7222, 7607, 8002, 8407, 8822, 9247, 9682, 10127, 10582

Offset: 0

N. J. A. Sloane

Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3^3.4^2 2D tiling (cf. A008706). - N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #13.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Reticular Chemistry Structure Resource (RCSR), The svk tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008706, A206399.
See A063489 for partial sums.
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.

lst={};Do[AppendTo[lst,5*n^2+2],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Join[{1}, 5 Range[46]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)

A010001(n)=5*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

G.f.: (1+x)*(1+3*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*5+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(10)/20*Pi*coth( Pi/5 *sqrt 10) = 1.2657655... - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A008527 Coordination sequence for body-centered tetragonal lattice.

Original entry on oeis.org

1, 10, 34, 74, 130, 202, 290, 394, 514, 650, 802, 970, 1154, 1354, 1570, 1802, 2050, 2314, 2594, 2890, 3202, 3530, 3874, 4234, 4610, 5002, 5410, 5834, 6274, 6730, 7202, 7690, 8194, 8714, 9250, 9802, 10370, 10954, 11554, 12170, 12802, 13450, 14114, 14794, 15490, 16202, 16930, 17674

Offset: 0

N. J. A. Sloane

Also sequence found by reading the segment (1, 10) together with the line from 10, in the direction 10, 34, ..., in the square spiral whose vertices are the generalized hexagonal numbers A000217. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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 (3,-3,1).

Apart from leading term, same as A108100.
Cf. A206399.
Cf. A016754 (SE), A054554 (NE), A054569 (SW), A053755 (NW), A033951 (S), A054552 (E), A054556 (N), A054567 (W) (Ulam spiral spokes).
A143839 (SSE) + A143855 (ESE) = A143838 (SSW) + A143856 (ENE) = A143854 (WSW) + A143861 (NNE) = A143859 (WNW) + A143860 (NNW) = even bisection = a(2n) = A010021(n).

Concatenation([1], List([1..40], n-> 2*(1+4*n^2) )); # G. C. Greubel, Nov 09 2019

[1] cat [2*(1 + 4*n^2): n in [1..50]]; // G. C. Greubel, Nov 09 2019

Maple
```
1, seq(8*k^2+2, k=1..50);
```

a[0]:= 1; a[n_]:= 8n^2 +2; Table[a[n], {n,0,50}] (* Alonso del Arte, Sep 06 2011 *)
LinearRecurrence[{3,-3,1},{1,10,34,74},50] (* Harvey P. Dale, Feb 13 2022 *)

vector(51, n, if(n==1,1, 2*(1+(2*n-2)^2)) ) \\ G. C. Greubel, Nov 09 2019

[1]+[2*(1+4*n^2) for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(0) = 1; a(n) = 8*n^2+2 for n>0.
From Gary W. Adamson, Dec 27 2007: (Start)
a(n) = (2n+1)^2 + (2n-1)^2 for n>0.
Binomial transform of [1, 9, 15, 1, -1, 1, -1, 1, ...]. (End)
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: (1+x)*(1+6*x+x^2)/(1-x)^3. (End)
From Bruce J. Nicholson, Jul 31 2019: (Start) Assume n>0.
a(n) = A016754(n) + A016754(n-1).
a(n) = 2 * A053755(n).
a(n) = A054554(n+1) + A054569(n+1).
a(n) = A033951(n) + A054552(n).
a(n) = A054556(n+1) + A054567(n+1). (End)
E.g.f.: -1 + 2*exp(x)*(1 + 2*x)^2. - Stefano Spezia, Aug 02 2019
Sum_{n>=0} 1/a(n) = 3/4+1/8*Pi*coth(Pi/2) = 1.178172.... - R. J. Mathar, May 07 2024

A010000 a(0) = 1, a(n) = n^2 + 2 for n > 0.

Original entry on oeis.org

1, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403

Offset: 0

N. J. A. Sloane

Least k such that A070864(k) = 2n-1. - Robert G. Wilson v and Benoit Cloitre, May 20 2002
With an offset of 3, beginning with 6 (deleting first two terms) n*(n+a(n)) + 1 is a cube = (n+1)^3: 1*(1+6) + 1 = 8, 2*(2+11) + 1 = 27, etc. - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003
For n >= 2, a(n) is the maximum element in the continued fraction for Sum_{k>=1} 1/n^(2^k) (for n=2 see A006464). - Benoit Cloitre, Jun 12 2007
Equals binomial transform of [1, 2, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008
Minimum Wiener index of 3-degenerate graphs with n+2 vertices.  A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices.  The extremal graphs are maximal 3-degenerate graphs with diameter at most 2. - Allan Bickle, Oct 14 2022
a(n-1) is the number of unit triangles enclosed by the triangular spiral drawn on a isometric grid of which the n-th side has length n. The picture in the link shows how the spiral is constructed. - Bob Andriesse, Feb 14 2023

			G.f. = 1 + 3*x + 6*x^2 + 11*x^3 + 18*x^4 + 27*x^5 + 38*x^6 + 51*x^7 + 66*x^8 + ...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bob Andriesse, Graphical representation of the first 21 segments of the spiral with grid.
Allan Bickle, Wiener indices of maximal k-degenerate graphs, International Journal of Mathematical Combinatorics 2 (2021) 68-79.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A070864. Apart from initial terms, same as A059100.
Cf. A206399.
Cf. A002061 (minimum Wiener index of 2-degenerate graphs).

a[1] = a[2] = 1; a[n_] := a[n] = 2 + a[n - a[n - 1]]; b = Table[0, {100}]; Do[c = (a[n] + 1)/2; If[c < 101 && b[[c]] == 0, b[[c]] = n], {n, 1, 10^4}]; b
Join[{1}, Range[50]^2 + 2] (* Bruno Berselli, Feb 08 2012 *)
a[ n_] := n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 05 2015 *)

a(n):=if n=0 then 1 else 2*n*sum((-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1),j,0,n)/(n+1)!; /* Vladimir Kruchinin, Jun 03 2013 */

{a(n) = n^2 + 2 - (n==0)}; /* Michael Somos, Aug 11 2009 */

a(n) = A000217(n-2) + A000217(n+1) for n > 0. - Jon Perry, Jul 23 2003
Euler transform of length 6 sequence [ 3, 0, 1, 0, 0, -1]. - Michael Somos, Aug 11 2009
From Michael Somos, Aug 11 2009: (Start)
G.f.: (1 + x^3) / (1 - x)^3.
a(n) = a(-n) for all n in Z. (End)
E.g.f.: (x*(x+1) + 2)*e^x - 1. - Gopinath A. R., Feb 14 2012
a(n) = (2*n/(n+1)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1), n > 0, a(0)=1. - Vladimir Kruchinin, Jun 03 2013
Sum_{n>=0} 1/a(n) = 3/4+1/4*sqrt(2)*Pi*coth(Pi*sqrt 2)= 1.8610281... - R. J. Mathar, May 07 2024

A005918 Number of points on surface of square pyramid: 3*n^2 + 2 (n>0).

Original entry on oeis.org

1, 5, 14, 29, 50, 77, 110, 149, 194, 245, 302, 365, 434, 509, 590, 677, 770, 869, 974, 1085, 1202, 1325, 1454, 1589, 1730, 1877, 2030, 2189, 2354, 2525, 2702, 2885, 3074, 3269, 3470, 3677, 3890, 4109, 4334, 4565, 4802, 5045, 5294, 5549, 5810, 6077, 6350, 6629

Offset: 0

N. J. A. Sloane

Also coordination sequence of the 5-connected (or bnn) net = hexagonal net X integers.
Also (except for initial term) numbers of the form 3n^2+2 that are not squares. All numbers 3n^2+2 are == 2 (mod 3), and hence not squares. - Cino Hilliard, Mar 01 2003, modified by Franklin T. Adams-Watters, Jun 27 2014
If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
Sums of three consecutive squares: (n - 2)^2 + (n - 1)^2 + n^2 for n > 1. - Keith Tyler, Aug 10 2010

			G.f. = 1 + 5*x + 14*x^2 + 29*x^3 + 50*x^4 + 77*x^5 + 110*x^6 + 149*x^7 + ...

H. S. M. Coxeter, Polyhedral numbers, in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. F. Wells, Three-Dimensional Nets and Polyhedra, Fig. 15.1 (e).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Branko Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #26.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From _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.
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 bnn tiling (or net).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A120328, A206399.

Partial sums give A063488.

A005918:=-(z+1)*(z**2+z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation.

Join[{1}, Table[Plus@@(Range[n, n + 2]^2), {n, 0, 49}]] (* Alonso del Arte, Oct 27 2012 *)
CoefficientList[Series[(1 - x^2) (1 - x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2014 *)
LinearRecurrence[{3,-3,1},{1,5,14,29},50] (* Harvey P. Dale, Dec 12 2015 *)

sq3nsqp2(n) = { for(x=1,n, y = 3*x*x+2; print1(y, ", ") ) }

{a(n) = 3*n^2 + 2 - (n==0)}; /* Michael Somos, Aug 07 2014 */

G.f.: (1 - x^2)*(1 - x^3)/(1 - x)^5 = (1+x)*(1+x+x^2)/(1-x)^3.
Euler transform of length 3 sequence [ 5, -1, -1]. - Michael Somos, Aug 07 2014
a(-n) = a(n) for all n in Z. - Michael Somos, Aug 07 2014
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3. - Colin Barker, Aug 07 2014
a(0) = 1; for n > 0, a(n) = A120328(n-1). - Doug Bell, Aug 18 2015
E.g.f.: (2+3*x+3*x^2)*exp(x)-1. - Robert Israel, Aug 18 2015
a(n) = A005448(n) + A005448(n+1), sum of 2 consecutive centered triangular numbers. - R. J. Mathar, Apr 28 2020
a(n) = (n - 1)^2 + n^2 + (n + 1)^2. - Charlie Marion, Aug 31 2021
From Amiram Eldar, Sep 14 2022: (Start)
Sum_{n>=0} 1/a(n) = coth(sqrt(2/3)*Pi)*Pi/(2*sqrt(6)) + 3/4.
Sum_{n>=0} (-1)^n/a(n) = cosech(sqrt(2/3)*Pi)*Pi/(2*sqrt(6)) + 3/4. (End)

A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

Original entry on oeis.org

1, 18, 66, 146, 258, 402, 578, 786, 1026, 1298, 1602, 1938, 2306, 2706, 3138, 3602, 4098, 4626, 5186, 5778, 6402, 7058, 7746, 8466, 9218, 10002, 10818, 11666, 12546, 13458, 14402, 15378, 16386, 17426, 18498, 19602, 20738, 21906, 23106, 24338, 25602, 26898

Offset: 0

N. J. A. Sloane, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

If Y_i (i=1,2,3) are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 66, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Milan Janjic, Two Enumerative Functions
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 (3,-3,1).

Cf. A206399. For the coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C_3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50). Cf. A137513.

[1] cat [16*n^2+2: n in [1..50]]; // Vincenzo Librandi, Feb 20 2012

Join[{1},Table[16n^2+2,{n,50}]] (* Harvey P. Dale, Oct 15 2012 *)

A010006(n)=16*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

a(0)=1, a(n) = 16*n^2 + 2, n >= 1.
G.f.: (1+x)*(1+14*x+x^2)/(1-x)^3.
G.f. for coordination sequence of C_n lattice: (1/(1-z)^n)*Sum_{i=0..n} binomial(2*n, 2*i)*z^i.
E.g.f.: (x*(x+1)*16+2)*e^x - 1. - Gopinath A. R., Feb 14 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=66, a(3)=146. - Harvey P. Dale, Oct 15 2012
G.f. for sequence with interpolated zeros: cosh(6*arctanh(x)) = (1/2)*( ((1 - x)/(1 + x))^3 + ((1 + x)/(1 - x))^3) = 1 + 18*x^2 + 66*x^4 + 146*x^6 + .... More generally, cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. Note that exp(t*arctanh(x)) is the e.g.f. for the Mittag_Leffler polynomials. See A137513. - Peter Bala, Apr 09 2017
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(2)/16*Pi*coth( Pi*sqrt(2)/4) = 1.095237238050... - R. J. Mathar, May 07 2024
a(n) = 2*A081585(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069129(n)+A069129(n+1). - R. J. Mathar, May 07 2024

A005914 Number of points on surface of hexagonal prism: 12*n^2 + 2 for n > 0 (coordination sequence for W(2)).

Original entry on oeis.org

1, 14, 50, 110, 194, 302, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810, 6350, 6914, 7502, 8114, 8750, 9410, 10094, 10802, 11534, 12290, 13070, 13874, 14702, 15554, 16430, 17330, 18254, 19202, 20174, 21170

Offset: 0

N. J. A. Sloane and Ralf W. Grosse-Kunstleve

For n >= 1, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n,n+1} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007
Equals binomial transform of [1, 13, 23, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 22 2008
First bisection of A005918. After 1, all terms are in A000408 (see Formula section). - Bruno Berselli, Feb 07 2012
Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 50, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Nov 02 2012
Unique sequence such that for all n > 0, n*a(1) + (n-1)*a(2) + (n-3)*a(3) + ... + 2*a(2) + a(1) = n^4. - Warren Breslow, Dec 12 2014

Gmelin Handbook of Inorganic and Organometallic Chemistry, 8th Ed., 1994, TYPIX search code (229) cI2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ovidiu Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, No. 2 (2014), pp. 91-100.
Ralf W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences, 1996.
Ralf 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, Enumerative Formulas for Some Functions on Finite Sets.
Claudio de J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq., Vol. 16 (2013), Article 13.5.7.
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.
Boon K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem., Vol. 24 (1985), pp. 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A005917.

Cf. A000408, A001082, A003154, A005918, A206399.

A005914:=-(z+1)*(z**2+10*z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation.

Table[If[n == 0, 1, 12*n^2 + 2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Join[{1},LinearRecurrence[{3,-3,1},{14,50,110},50]] (* Harvey P. Dale, Oct 09 2012 *)

a(n)=12*n^2+2 \\ Charles R Greathouse IV, Jan 31 2012

G.f.: (1+x)*(1+10*x+x^2)/(1-x)^3. - Simon Plouffe (see MAPLE line)
a(n) = (2n-1)^2 + (2n)^2 + (2n+1)^2 for n > 0. - Bruno Berselli, Jan 30 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=14, a(2)=50, a(3)=110. - Harvey P. Dale, Oct 09 2012
E.g.f.: exp(x)*(12*x^2 + 12*x + 2) - 1. - Alois P. Heinz, Sep 10 2013
From Bruce J. Nicholson, Jan 19 2019: (Start)
Sum_{i=1..n} a(i) = A005917(n+1).
a(n) = A003154(n) + A003154(n+1). (End)
From Amiram Eldar, Jan 27 2022: (Start)
Sum_{n>=0} 1/a(n) = ((Pi/sqrt(6))*coth(Pi/sqrt(6)) + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = ((Pi/sqrt(6))*cosech(Pi/sqrt(6)) + 3)/4. (End)

cp's OEIS Frontend

A005897 a(n) = 6*n^2 + 2 for n > 0, a(0)=1.

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A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

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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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A005901 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.

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A010001 a(0) = 1, a(n) = 5*n^2 + 2 for n>0.

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A008527 Coordination sequence for body-centered tetragonal lattice.

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