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Search: id:A001845
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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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+0
27
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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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Number of points in simple cubic lattice at 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-subests of X intersecting each Y_i (i=1,2,3). - Milan R. Janjic (agnus(AT)blic.net), 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 (qntmpkt(AT)yahoo.com), Jul 19 2008
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REFERENCES
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S. 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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
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
Milan Janjic, Two Enumerative Functions
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for crystal ball sequences
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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 (alex(AT)kolmogorov.com), May 23 2006
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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 by S. Plouffe in his 1992 dissertation.]
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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. A014820.
Cf. A013609.
Adjacent sequences: A001842 A001843 A001844 this_sequence A001846 A001847 A001848
Sequence in context: A033814 A118395 A118396 this_sequence A127765 A056685 A001296
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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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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