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 A005918 Number of points on surface of square pyramid: 3*n^2 + 2 (n>0). (Formerly M3843) 17
 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. 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). 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) 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 * A321178 A256666 A319007 Adjacent sequences: A005915 A005916 A005917 * A005919 A005920 A005921 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified September 27 10:59 EDT 2023. Contains 365688 sequences. (Running on oeis4.)