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%I M4904 N2102 #68 Mar 23 2024 08:22:41
%S 1,13,85,377,1289,3653,8989,19825,40081,75517,134245,227305,369305,
%T 579125,880685,1303777,1884961,2668525,3707509,5064793,6814249,
%U 9041957,11847485,15345233,19665841,24957661,31388293,39146185,48442297,59511829,72616013,88043969
%N Crystal ball sequence for 6-dimensional cubic lattice.
%C Number of nodes of degree 12 in virtual, optimal chordal graphs of diameter d(G)=n. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
%C Equals binomial transform of [1, 12, 60, 160, 240, 192, 64, 0, 0, 0, ...] where (1, 12, 60, 160, 240, 192, 64) = row 6 of the Chebyshev triangle A013609. - _Gary W. Adamson_, Jul 19 2008
%C a(n) is the number of points in Z^6 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 6 from any given point. - _Shel Kaphan_, Jan 02 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 231.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001848/b001848.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, <a href="http://www.cecm.sfu.ca/~choi/paper/lrh.pdf">A local Riemann hypothesis, I</a> pages 16 and 17
%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%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 R. G. Stanton and D. D. Cowan, <a href="http://dx.doi.org/10.1137/1012049">Note on a "square" functional equation</a>, SIAM Rev., 12 (1970), 277-279.
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: (1+x)^6 /(1-x)^7.
%F a(n) = (4/45)*n^6 + (4/15)*n^5 + (14/9)*n^4 + (8/3)*n^3 + (196/45)*n^2 + (46/15)*n + 1. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
%F a(n) = Sum_{k=0..min(6,n)} 2^k * binomial(6,k)* binomial(n,k). See Bump et al. - _Tom Copeland_, Sep 05 2014
%F Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - 37/60 = log(2) - (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6). - _Peter Bala_, Mar 23 2024
%p for n from 1 to k do eval(4/45*n^6+4/15*n^5+14/9*n^4+8/3*n^3+196/45*n^2+46/15*n+1); od;
%p A001848:=-(z+1)**6/(z-1)**7; # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation
%t CoefficientList[Series[-(z + 1)^6/(z - 1)^7, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011 *)
%Y Cf. A001847, A013609.
%Y Cf. A240876.
%Y Row/column 6 of A008288.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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