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A002664
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a(n) = 2^n - C(n,0)- ... - C(n,4).
(Formerly M4395 N1851)
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19
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0, 0, 0, 0, 0, 1, 7, 29, 93, 256, 638, 1486, 3302, 7099, 14913, 30827, 63019, 127858, 258096, 519252, 1042380, 2089605, 4185195, 8377705, 16764265, 33539156, 67090962, 134196874, 268411298, 536843071, 1073709893
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OFFSET
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0,7
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COMMENTS
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Starting with "1" = eigensequence of a triangle with binomial C(n,5):
(1, 6, 21, 56, ...) as the left border and the rest 1's. (End)
Starting (0, 0, 0, 0, 1, 7, 29, ...), this is the binomial transform of (0, 0, 0, 0, 1, 2, 2, 2, ...). Starting (1, 7, 29, ...), this is the binomial transform of (1, 6, 16, 26, 31, 32, 32, 32, ...). - Gary W. Adamson, Jul 28 2015
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REFERENCES
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J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, Chapter 3, pp. 76-79.
J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: x^5/((1-2*x)*(1-x)^5).
a(n) = Sum_{k=0..n} C(n, k+5) = Sum_{k=5..n} C(n, k); a(n) = 2a(n-1) + C(n-1, 4). - Paul Barry, Aug 23 2004
a(n) = 2^n - n^4/24 + n^3/12 - 11*n^2/24 - 7*n/12 - 1. - Bruno Berselli, May 19 2011 [Robinson (1985) gives an alternative version of this formula, for a different offset. - N. J. A. Sloane, Oct 20 2015]
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MAPLE
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a:=n->sum(binomial(n+1, 2*j), j=3..n+1): seq(a(n), n=0..30); # Zerinvary Lajos, May 12 2007
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MATHEMATICA
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a=1; lst={}; s1=s2=s3=s4=s5=0; Do[s1+=a; s2+=s1; s3+=s2; s4+=s3; s5+=s4; AppendTo[lst, s5]; a=a*2, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Table[Sum[ Binomial[n, k + 5], {k, 0, n}], {n, 0, 30}] (* Zerinvary Lajos, Jul 08 2009 *)
Table[2^n-Total[Binomial[n, Range[0, 4]]], {n, 0, 30}] (* or *) LinearRecurrence[ {7, -20, 30, -25, 11, -2}, {0, 0, 0, 0, 0, 1}, 40] (* Harvey P. Dale, Sep 03 2016 *)
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PROG
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(Magma) [2^n-n^4/24+n^3/12-11*n^2/24-7*n/12-1: n in [0..35]]; // Vincenzo Librandi, May 20 2011
(Haskell)
a002664 n = a002664_list !! n
a002664_list = map (sum . drop 5) a007318_tabl
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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