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A039823
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Ceiling[ (n^2+n+2)/4 ].
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0
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1, 2, 4, 6, 8, 11, 15, 19, 23, 28, 34, 40, 46, 53, 61, 69, 77, 86, 96, 106, 116, 127, 139, 151, 163, 176, 190, 204, 218, 233, 249, 265, 281, 298, 316, 334, 352, 371, 391, 411, 431, 452, 474, 496, 518, 541, 565, 589, 613, 638, 664, 690, 716, 743, 771, 799, 827
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Equals number of different coefficient values in expansion of Product (1+q^1+...+q^i), i=1 to n. Proof by Lawrence Sze: The Gaussian polynomial Prod[k=1..n, Sum[j=0..k, q^j]] is the q-version of n! and strictly unimodal with constant term 1. It has degree Sum[k=1..n, k]=n(n+1)/2 and thus n(n+1)/2+1 nonzero terms.
a(n) is equivalently the number of different absolute values obtained when summing the first n integers with all possible 2^n sign combinations. [From Olivier GERARD (olivier.gerard(AT)gmail.com), Mar 22 2010]
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FORMULA
| [ C(n+1, 2)/2 ] + 1.
G.f.: x(x^4-2x^3+2x^2-x+1)/[(1+x^2)(1-x)^3].
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EXAMPLE
| Possible absolute values of sums of consecutive integers with any sign combination for n = 4 and n=5 are {0, 2, 4, 6, 8, 10} and {1, 3, 5, 7, 9, 11, 13, 15} respectively. [From Olivier GERARD (olivier.gerard(AT)gmail.com), Mar 22 2010]
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MATHEMATICA
| Table[Floor[((n*(n+1)+2)/2+1)/2], {n, 5!}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 26 2010]
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CROSSREFS
| Equals A011848(n+1) + 1.
Cf. A000125, A063865. [From Olivier GERARD (olivier.gerard(AT)gmail.com), Mar 22 2010]
Sequence in context: A194224 A194252 A205727 * A079972 A164144 A071241
Adjacent sequences: A039820 A039821 A039822 * A039824 A039825 A039826
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KEYWORD
| nonn
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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EXTENSIONS
| Edited by Ralf Stephan, Nov 15 2004
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