cp's OEIS Frontend

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

%I A005914 M4931 #93 Jan 27 2022 03:08:27
%S A005914 1,14,50,110,194,302,434,590,770,974,1202,1454,1730,2030,2354,2702,
%T A005914 3074,3470,3890,4334,4802,5294,5810,6350,6914,7502,8114,8750,9410,
%U A005914 10094,10802,11534,12290,13070,13874,14702,15554,16430,17330,18254,19202,20174,21170
%N A005914 Number of points on surface of hexagonal prism: 12*n^2 + 2 for n > 0 (coordination sequence for W(2)).
%C A005914 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
%C A005914 Equals binomial transform of [1, 13, 23, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Apr 22 2008
%C A005914 First bisection of A005918. After 1, all terms are in A000408 (see Formula section). - _Bruno Berselli_, Feb 07 2012
%C A005914 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
%C A005914 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
%D A005914 Gmelin Handbook of Inorganic and Organometallic Chemistry, 8th Ed., 1994, TYPIX search code (229) cI2.
%D A005914 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005914 Vincenzo Librandi, <a href="/A005914/b005914.txt">Table of n, a(n) for n = 0..10000</a>
%H A005914 Ovidiu Bagdasar, <a href="https://derby.openrepository.com/handle/10545/583372">On Some Functions Involving the lcm and gcd of Integer Tuples</a>, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, No. 2 (2014), pp. 91-100.
%H A005914 Ralf W. Grosse-Kunstleve, <a href="/A005897/a005897.html">Coordination Sequences and Encyclopedia of Integer Sequences</a>, 1996.
%H A005914 Ralf W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, <a href="http://neilsloane.com/doc/ac96cs/">Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites</a>, Acta Cryst., A52 (1996), pp. 879-889.
%H A005914 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A005914 Claudio de J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq., Vol. 16 (2013), Article 13.5.7.
%H A005914 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H A005914 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005914 Boon K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem., Vol. 24 (1985), pp. 4545-4558.
%H A005914 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A005914 G.f.: (1+x)*(1+10*x+x^2)/(1-x)^3. - _Simon Plouffe_ (see MAPLE line)
%F A005914 a(n) = (2n-1)^2 + (2n)^2 + (2n+1)^2 for n > 0. - _Bruno Berselli_, Jan 30 2012
%F A005914 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
%F A005914 E.g.f.: exp(x)*(12*x^2 + 12*x + 2) - 1. - _Alois P. Heinz_, Sep 10 2013
%F A005914 From _Bruce J. Nicholson_, Jan 19 2019: (Start)
%F A005914 Sum_{i=1..n} a(i) = A005917(n+1).
%F A005914 a(n) = A003154(n) + A003154(n+1). (End)
%F A005914 From _Amiram Eldar_, Jan 27 2022: (Start)
%F A005914 Sum_{n>=0} 1/a(n) = ((Pi/sqrt(6))*coth(Pi/sqrt(6)) + 3)/4.
%F A005914 Sum_{n>=0} (-1)^n/a(n) = ((Pi/sqrt(6))*cosech(Pi/sqrt(6)) + 3)/4. (End)
%p A005914 A005914:=-(z+1)*(z**2+10*z+1)/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation.
%t A005914 Table[If[n == 0, 1, 12*n^2 + 2], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011 *)
%t A005914 Join[{1},LinearRecurrence[{3,-3,1},{14,50,110},50]] (* _Harvey P. Dale_, Oct 09 2012 *)
%o A005914 (PARI) a(n)=12*n^2+2 \\ _Charles R Greathouse IV_, Jan 31 2012
%Y A005914 First differences of A005917.
%Y A005914 Cf. A000408, A001082, A003154, A005918, A206399.
%K A005914 nonn,easy,nice
%O A005914 0,2
%A A005914 _N. J. A. Sloane_ and _Ralf W. Grosse-Kunstleve_