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Search: id:A099193
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| A099193 |
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Figurate numbers based on the 7-dimensional regular convex polytope called the 7-dimensional cross-polytope, or 7-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 3, 4}. It is the dual of the 7-dimensional hypercube. |
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+0
7
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| 0, 1, 14, 99, 476, 1765, 5418, 14407, 34232, 74313, 149830, 284075, 511380, 880685, 1459810, 2340495, 3644272, 5529233, 8197758, 11905267, 16970060
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Kim asserts that every nonnegative integer can be represented by the sum of no more than 21 of these 7-crosspolytope numbers.
Starting with "1" = binomial transform of [1, 13, 72, 220, 400, 432, 256, 0, 0, 0,...], where (1, 13, 72, 220, 400, 432, 256) = row 7 of the Chebyshev triangle A081277. Also = row 7 of the array in A142978. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008
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REFERENCES
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H. S. M. Coxeter, Regular Polytopes, New York: Dover, 1973.
J. V. Post, "4-Dimensional Jonathan numbers: polytope numbers and Centered polytope numbers of Higher Than 3 Dimensions", Draft 1.5 of 9 a.m., 12 March 2004, circulated by e-mail.
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LINKS
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H. K. Kim, "On Regular polytope numbers".
J. V. Post, Table of polytope numbers, Sorted, Through 1,000,000.
J. V. Post, Math Pages.
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FORMULA
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a(n) = 7-crosspolytope(n) = n*(4*n^6 + 70*n^4 + 196*n^2 + 45 )/315.
G.f.: x*(1+x)^6/(1-x)^8. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 18 2009]
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EXAMPLE
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a(3) = 99 because 3*(4*3^6 + 70*3^4 + 196*3^2 + 45 )/315 = 99.
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CROSSREFS
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Cf. A000332, A014820, A005900.
Cf. A142978, A081277.
Sequence in context: A008415 A003206 A101376 this_sequence A086951 A041370 A055913
Adjacent sequences: A099190 A099191 A099192 this_sequence A099194 A099195 A099196
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 16 2004
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