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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

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

References

• 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).

Links

• 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).

Crossrefs

Cf. A120328, A206399.

Partial sums give A063488.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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)