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A005918 Number of points on surface of square pyramid: 3*n^2 + 2 (n>0).
(Formerly M3843)
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

REFERENCES

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 #26.

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

Milan Janjic, Two Enumerative Functions

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013

M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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).

FORMULA

G.f.: (1 - x^2)*(1 - x^3)/(1 - x)^5.

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

EXAMPLE

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

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 *)

PROG

(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 */

CROSSREFS

Cf. A120328, A206399.

Partial sums give A063488.

Sequence in context: A161437 A301681 A047801 * A256666 A319007 A211651

Adjacent sequences:  A005915 A005916 A005917 * A005919 A005920 A005921

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 18 23:07 EDT 2018. Contains 316327 sequences. (Running on oeis4.)