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A027306
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2^(n - 1) + (1 + ( - 1)^n)/4*binomial(n, n/2).
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8
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1, 1, 3, 4, 11, 16, 42, 64, 163, 256, 638, 1024, 2510, 4096, 9908, 16384, 39203, 65536, 155382, 262144, 616666, 1048576, 2449868, 4194304, 9740686, 16777216, 38754732, 67108864, 154276028, 268435456, 614429672, 1073741824, 2448023843
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OFFSET
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0,3
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COMMENTS
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Equals Sum_{k=0..[n/2]} C(n,k).
Inverse binomial transform of A027914 . Hankel transform (see A001906 for definition) is {1, 2, 3, 4, ..., n, ...} . - Philippe DELEHAM, Jul 21 2005
Number of walks of length n on a line that starts at the origin and ends at or above 0 -Benjamin Phillabaum (bphillab (AT) gmail.com), Mar 05 2011
Number of binary integers (i.e. with a leading 1 bit) of length n+1 which have a majority of 1-bits. E.g. for n+1=4: (1011, 1101, 1110, 1111) a(3)=4. - Toby Gottfried, Dec 11 2011
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REFERENCES
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A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.1.6)
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LINKS
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Table of n, a(n) for n=0..32.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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FORMULA
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Odd terms are 2^(n-1). Also a(2n) - 2^(2n-1) is given by A001700. a(n) = 2^n+mod(n, 2)*C(n, (n-1)/2).
E.g.f.: (exp(2x)+I_0(2x))/2.
O.g.f.: 2*x/(1-2*x)/(1+2*x-((1+2*x)*(1-2*x))^(1/2)). - Vladeta Jovovic, Apr 27 2003
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MAPLE
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a:= proc(n) option remember; if n=0 then 1 else add(binomial (n, j), j=0..n/2) fi end: seq (a(n), n=0..32); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 29 2009]
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MATHEMATICA
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Table[Sum[Binomial[n, k], {k, 0, Floor[n/2]}], {n, 1, 35}]
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PROG
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(PARI) a(n)=if(n<0, 0, (2^n+if(n%2, 0, binomial(n, n/2)))/2)
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CROSSREFS
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a(n) = Sum{(k+1)T(n, m-k)}, 0<=k<=[ (n+1)/2 ], T given by A008315.
Sequence in context: A001641 A007382 A127804 * A026676 A142870 A143680
Adjacent sequences: A027303 A027304 A027305 * A027307 A027308 A027309
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KEYWORD
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nonn,easy,walk,changed
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Better description from Robert G. Wilson v, Aug 30 2000 and from Yong Kong (ykong(AT)curagen.com), Dec 28 2000
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STATUS
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approved
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