A005918 - OEIS

%I M3843 #102 Sep 14 2022 02:00:51

%S 1,5,14,29,50,77,110,149,194,245,302,365,434,509,590,677,770,869,974,

%T 1085,1202,1325,1454,1589,1730,1877,2030,2189,2354,2525,2702,2885,

%U 3074,3269,3470,3677,3890,4109,4334,4565,4802,5045,5294,5549,5810,6077,6350,6629

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

%C Also coordination sequence of the 5-connected (or bnn) net = hexagonal net X integers.

%C 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

%C 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

%C Sums of three consecutive squares: (n - 2)^2 + (n - 1)^2 + n^2 for n > 1. - _Keith Tyler_, Aug 10 2010

%D H. S. M. Coxeter, Polyhedral numbers, in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D A. F. Wells, Three-Dimensional Nets and Polyhedra, Fig. 15.1 (e).

%H Vincenzo Librandi, <a href="/A005918/b005918.txt">Table of n, a(n) for n = 0..1000</a>

%H Branko Grünbaum, <a href="http://faculty.washington.edu/moishe/branko/BG199.Uniform%20Tilings%20of%203-Space.pdf">Uniform tilings of 3-space</a>, Geombinatorics, 4 (1994), 49-56. See tiling #26.

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.

%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550, 2013. - From _N. J. A. Sloane_, Feb 13 2013

%H Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014), Article 14.3.5.

%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/bnn">The bnn tiling (or net)</a>.

%H B. K. Teo and N. J. A. Sloane, <a href="http://neilsloane.com/doc/magic1/magic1.html">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985),4545-4558.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (1 - x^2)*(1 - x^3)/(1 - x)^5 = (1+x)*(1+x+x^2)/(1-x)^3.

%F Euler transform of length 3 sequence [ 5, -1, -1]. - _Michael Somos_, Aug 07 2014

%F a(-n) = a(n) for all n in Z. - _Michael Somos_, Aug 07 2014

%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3. - _Colin Barker_, Aug 07 2014

%F a(0) = 1; for n > 0, a(n) = A120328(n-1). - _Doug Bell_, Aug 18 2015

%F E.g.f.: (2+3*x+3*x^2)*exp(x)-1. - _Robert Israel_, Aug 18 2015

%F a(n) = A005448(n) + A005448(n+1), sum of 2 consecutive centered triangular numbers. - _R. J. Mathar_, Apr 28 2020

%F a(n) = (n - 1)^2 + n^2 + (n + 1)^2. - _Charlie Marion_, Aug 31 2021

%F From _Amiram Eldar_, Sep 14 2022: (Start)

%F Sum_{n>=0} 1/a(n) = coth(sqrt(2/3)*Pi)*Pi/(2*sqrt(6)) + 3/4.

%F Sum_{n>=0} (-1)^n/a(n) = cosech(sqrt(2/3)*Pi)*Pi/(2*sqrt(6)) + 3/4. (End)

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

%p A005918:=-(z+1)*(z**2+z+1)/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation.

%t Join[{1}, Table[Plus@@(Range[n, n + 2]^2), {n, 0, 49}]] (* _Alonso del Arte_, Oct 27 2012 *)

%t CoefficientList[Series[(1 - x^2) (1 - x^3)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 07 2014 *)

%t LinearRecurrence[{3,-3,1},{1,5,14,29},50] (* _Harvey P. Dale_, Dec 12 2015 *)

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

%o (PARI) {a(n) = 3*n^2 + 2 - (n==0)}; /* _Michael Somos_, Aug 07 2014 */

%Y Cf. A120328, A206399.

%Y Partial sums give A063488.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_