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

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

Author

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

Keywords

• nonn
• easy
• nice

Comments

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

References

• 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).

Links

• 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).

Crossrefs

First differences of A005917.

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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)