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Search: id:A001846
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| A001846 |
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Centered 4-dimensional orthoplex numbers (crystal ball sequence for 4-dimensional cubic lattice).
(Formerly M4622 N1974)
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+0
5
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| 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
(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 nodes 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 a (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 R. Janjic (agnus(AT)blic.net), 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 (qntmpkt(AT)yahoo.com), Jul 19 2008
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
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
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.
Index entries for crystal ball sequences
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FORMULA
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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 (jvospost3(AT)gmail.com), Mar 15 2006
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EXAMPLE
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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
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MAPLE
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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 S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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Cf. A001847, A008412, A059100.
Cf. A013609.
Sequence in context: A000437 A095809 A018836 this_sequence A034441 A056243 A083584
Adjacent sequences: A001843 A001844 A001845 this_sequence A001847 A001848 A001849
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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