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A001846 Centered 4-dimensional orthoplex numbers (crystal ball sequence for 4-dimensional cubic lattice).
(Formerly M4622 N1974)
10
1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, 8361, 11969, 16641, 22569, 29961, 39041, 50049, 63241, 78889, 97281, 118721, 143529, 172041, 204609, 241601, 283401, 330409, 383041, 441729, 506921, 579081, 658689, 746241, 842249, 947241, 1061761, 1186369 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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

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

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.

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, A local Riemann hypothesis, I pages 16 and 17

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).

Milan Janjic, Two Enumerative Functions

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

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.

R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.

Ben Thurston, Illustration of the first four clusters of points in two dimensions

Index entries for crystal ball sequences

FORMULA

G.f.: (1+x)^4 /(1-x)^5.

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

a(n) = SUM[i=0..n] A008412(i); a(n) = SUM[i=0..n] (8*i)*(i^2+2)/3; a(n) = SUM[i=0..n] (8*i)*(A059100(i))/3. - Jonathan Vos Post, Mar 15 2006

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

EXAMPLE

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.

MAPLE

for n from 1 to k do eval((2*n^4+4*n^3+10*n^2+8*n+3)/3) od;

A001846:=-(z+1)**4/(z-1)**5; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

MATHEMATICA

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 *)

CROSSREFS

Cf. A005408, A001844, A001845, A001847, A059100, A013609.

First differences are A008412.

Cf. A240876.

Sequence in context: A274323 A297740 A297741 * A271663 A034441 A201275

Adjacent sequences:  A001843 A001844 A001845 * A001847 A001848 A001849

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 17 23:47 EST 2018. Contains 299297 sequences. (Running on oeis4.)