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A001845
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Centered octahedral numbers (crystal ball sequence for cubic lattice).
(Formerly M4384 N1844)
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32
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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
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OFFSET
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0,2
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COMMENTS
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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]
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
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).
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
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
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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).
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FORMULA
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G.f.: (1+x)^3 /(1-x)^4. 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
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MAPLE
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(1/3)*(2*n+1)*(2*n^2+2*n+3);
A001845:=(z+1)**3/(z-1)**4; [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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Table[(4*n^3-6*n^2+8*n-3)/3, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)
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PROG
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(PARI) a(n)=(2*n+1)*(2*n^2+2*n+3)/3 \\ Charles R Greathouse IV, Dec 06 2011
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CROSSREFS
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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.
Sequence in context: A118396 A193375 A185787 * A127765 A155305 A155290
Adjacent sequences: A001842 A001843 A001844 * A001846 A001847 A001848
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jul 17 2000
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STATUS
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approved
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