|
%I M4622 N1974 #111 Mar 23 2024 08:19:44
%S 1,9,41,129,321,681,1289,2241,3649,5641,8361,11969,16641,22569,29961,
%T 39041,50049,63241,78889,97281,118721,143529,172041,204609,241601,
%U 283401,330409,383041,441729,506921,579081,658689,746241,842249,947241,1061761,1186369
%N Centered 4-dimensional orthoplex numbers (crystal ball sequence for 4-dimensional cubic lattice).
%C a(n) is the number of points in the Z^4 lattice that are at distance at most n from the origin in the adjacency graph. - _N. J. A. Sloane_, Feb 19 2013
%C Number of nodes of degree 8 in virtual, optimal, chordal graphs of diameter d(G)=n. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 07 2002
%C If Y_i (i=1,2,3,4) are 2-blocks of an (n+4)-set X then a(n-4) is the number of 8-subsets of X intersecting each Y_i (i=1,2,3,4). - _Milan Janjic_, Oct 28 2007
%C Equals binomial transform of [1, 8, 24, 32, 16, 0, 0, 0, ...] where (1, 8, 24, 32, 16) = row 4 of the Chebyshev triangle A013609. - _Gary W. Adamson_, Jul 19 2008
%C Comment from Ben Thurston, Feb 18 2013: In the plane, if you make a picture by taking one unit step in each of the basic 8 directions from a central dot, then from each of those going one unit step in each of the eight directions, ... (see illustration), it appears that the number of dots in the picture after n steps is equal to a(n). Response from _N. J. A. Sloane_, Feb 19 2013: This is correct, and follows from the fact that the Z-module Z[1,i,(+-1+i)/sqrt(2)] is essentially a copy of the Z^4 lattice.
%C a(n) = D(4,n) where D are the Delannoy numbers (A008288). As such, a(n) gives the number of grid paths from (0,0) to (4,n) using steps that move one unit north, east, or northeast. - _Jack W Grahl_, Feb 15 2021
%C The first comment above can be re-expressed and generalized as follows: a(n) is the number of points in Z^4 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 <= 4 from any given point. - _Shel Kaphan_, Jan 02 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%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="/A001846/b001846.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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>. [Broken link; <a href="https://web.archive.org/web/20110204173116/http://www.pmfbl.org:80/janjic/">WayBackMachine archive</a>]
%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%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 Ben Thurston, <a href="/A001846/a001846.png">Illustration of the first four clusters of points in two dimensions</a>
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (1+x)^4 /(1-x)^5.
%F a(n) = (2*n^4 + 4*n^3 + 10*n^2 + 8*n + 3)/3. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 07 2002
%F From _Jonathan Vos Post_, Mar 15 2006: (Start)
%F a(n) = Sum_{i=0..n} A008412(i);
%F a(n) = Sum_{i=0..n} 8*i*(i^2 + 2)/3;
%F a(n) = Sum_{i=0..n} 8*i*(A059100(i))/3. (End)
%F a(n) = Sum_{k=0..min(4,n)} 2^k * binomial(4,k)* binomial(n,k). See Bump et al. - _Tom Copeland_, Sep 05 2014
%F E.g.f.: exp(x)*(3 + 24*x + 36*x^2 + 16*x^3 + 2*x^4)/3. - _Stefano Spezia_, Mar 14 2024
%F Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - 7/12 = log(2) - (1 - 1/2 + 1/3 - 1/4). - _Peter Bala_, Mar 23 2024
%e a(6)=1289: (2*6^4 + 4*6^3 + 10*6^2 + 8*6 + 3) / 3 = (2592 + 864 + 360 + 48 + 3) / 3 = 3867 / 3 = 1289.
%p for n from 1 to k do eval((2*n^4+4*n^3+10*n^2+8*n+3)/3) od;
%p A001846:=-(z+1)**4/(z-1)**5; # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation
%t CoefficientList[Series[(-z^4-4 z^3-6 z^2-4 z-1)/(z-1)^5, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011 *)
%Y Cf. A005408, A001844, A001845, A001847, A059100, A013609.
%Y First differences are A008412.
%Y Cf. A240876.
%Y Row/column 4 of A008288.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
|
