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Search: id:A055417
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| A055417 |
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Number of points in N^n of norm <= 2. |
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+0
1
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| 1, 3, 6, 11, 20, 36, 63, 106, 171, 265, 396, 573, 806, 1106, 1485, 1956, 2533, 3231, 4066, 5055, 6216, 7568, 9131, 10926, 12975, 15301, 17928, 20881, 24186, 27870, 31961, 36488, 41481, 46971, 52990, 59571, 66748, 74556, 83031, 92210, 102131
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Equals binomial transform of [1, 0, 2, -1, 2, -1, 1, -1, 1, -1, 1,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 12 2009]
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FORMULA
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a(n) = (n^3-3*n^2+14*n+24)*(n+1)/24. Proof: The coordinates of such a point are a permutation of one of the vectors (0, ..., 0), (0, ..., 0, 1), (0, ..., 0, 2), (0, ..., 0, 1, 1), (0, ..., 0, 1, 1, 1), or (0, ..., 0, 1, 1, 1, 1), so the number of points is 1+n+n+binomial(n, 2)+binomial(n, 3)+binomial(n, 4). - Formula conjectured by frank.ellermann(AT)t-online.de, Mar 16 2002 and explained by Michael Somos, Apr 25 2003
G.f.: (1-2x+x^2+x^3)/(1-x)^5. - Michael Somos, Apr 25 2003
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EXAMPLE
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{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), (0, 1, 1), (0, 2, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1), (2, 0, 0)} are all the points in N^3 of norm<=2 so a(3)=11.
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PROGRAM
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(PARI) a(n)=(n^3-3*n^2+14*n+24)*(n+1)/24
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CROSSREFS
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Sequence in context: A001911 A020957 A116365 this_sequence A018918 A077855 A054887
Adjacent sequences: A055414 A055415 A055416 this_sequence A055418 A055419 A055420
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson (davidwwilson(AT)comcast.net)
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