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A001845 Centered octahedral numbers (crystal ball sequence for cubic lattice).
(Formerly M4384 N1844)
37
1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, 3303, 4089, 4991, 6017, 7175, 8473, 9919, 11521, 13287, 15225, 17343, 19649, 22151, 24857, 27775, 30913, 34279, 37881, 41727, 45825, 50183, 54809, 59711, 64897, 70375, 76153, 82239 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of points in simple cubic lattice at most n steps from origin.

If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 6-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007

Equals binomial transform of [1, 6, 12, 8, 0, 0, 0,...] where (1, 6, 12, 8) = row 3 of the Chebyshev triangle A013609. - Gary W. Adamson, Jul 19 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-2)= -coeff(charpoly(A,x),x^(n-3)). [Milan Janjic, Jan 26 2010]

a(n) = A005408(n) * A097080(n-1) / 3. - Reinhard Zumkeller, Dec 15 2013

a(n) = D(3,n) where D are the Delannoy numbers (A008288). As such, a(n) counts the number of grid paths from (0,0) to (3,n) using steps that move one unit north, east, or northeast. - David Eppstein, Sep 07 2014

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 (Abstract, pdf, ps).

Milan Janjic, Two Enumerative Functions

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).

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.

Eric Weisstein's World of Mathematics, Octahedral Number

Index entries for crystal ball sequences

Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: (1+x)^3 /(1-x)^4. [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.]

a(n) = (2*n+1)*(2*n^2+2*n+3)/3.

First differences of A014820(n). - Alexander Adamchuk, May 23 2006

a(n) = a(n-1) +4*n^2+2, a(0)=1. - Vincenzo Librandi, Mar 27 2011

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4), with a(0)=1, a(1)=7, a(2)=25, a(3)=63. - Harvey P. Dale, Jun 05 2013

a(n)= sum(k=0..min(3,n), 2^k * binomial(3,k)* binomial(n,k) ). See Bump et al. - Tom Copeland, Sep 05 2014

MATHEMATICA

Table[(4*n^3-6*n^2+8*n-3)/3, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)

LinearRecurrence[{4, -6, 4, -1}, {1, 7, 25, 63}, 40] (* Harvey P. Dale, Jun 05 2013 *)

PROG

(PARI) a(n)=(2*n+1)*(2*n^2+2*n+3)/3 \\ Charles R Greathouse IV, Dec 06 2011

(Haskell)

a001845 n = (2 * n + 1) * (2 * n ^ 2 + 2 * n + 3) `div` 3

-- Reinhard Zumkeller, Dec 15 2013

CROSSREFS

Sums of 2 consecutive terms give A008412.

(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Partial sums of A005899.

Cf. A001846, A001847, A001848, etc., A014820, A013609.

Cf. A240876.

Sequence in context: A118396 A193375 A185787 * A127765 A155305 A155290

Adjacent sequences:  A001842 A001843 A001844 * A001846 A001847 A001848

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 17 2000

STATUS

approved

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Last modified November 21 19:13 EST 2014. Contains 249784 sequences.