A001848 Crystal ball sequence for 6-dimensional cubic lattice.

1, 13, 85, 377, 1289, 3653, 8989, 19825, 40081, 75517, 134245, 227305, 369305, 579125, 880685, 1303777, 1884961, 2668525, 3707509, 5064793, 6814249, 9041957, 11847485, 15345233, 19665841, 24957661, 31388293, 39146185, 48442297, 59511829, 72616013, 88043969

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

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

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 231.
• 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).
• 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
• R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
• Index entries for crystal ball sequences
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A001847, A013609.

Cf. A240876.

Row/column 6 of A008288.

Programs

Maple

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;
A001848:=-(z+1)**6/(z-1)**7; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[-(z + 1)^6/(z - 1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

Formula

G.f.: (1+x)^6 /(1-x)^7.

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

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

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