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A005126
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a(n) = 2^n + n + 1.
(Formerly M1061)
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23
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2, 4, 7, 12, 21, 38, 71, 136, 265, 522, 1035, 2060, 4109, 8206, 16399, 32784, 65553, 131090, 262163, 524308, 1048597, 2097174, 4194327, 8388632, 16777241, 33554458, 67108891, 134217756, 268435485, 536870942, 1073741855, 2147483680, 4294967329, 8589934626
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OFFSET
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0,1
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COMMENTS
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Binomial transform of (1, 1, 1, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Jul 20 2007
Binomial transform of a(n) starts: 2, 6, 17, 47, 129, 355, 985, 2763, 7841, 22499, 65193, 190459, ... - Wesley Ivan Hurt, Oct 28 2014
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REFERENCES
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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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Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 921
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
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FORMULA
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G.f.: (2-4*x+x^2)/((1-2*x)*(1-x)^2). - Simon Plouffe
E.g.f.: exp(x)*(exp(x)+1+x) = U(0) where U(k) = 1 + x/(2^k - 2^k/(x + 1 - x^2*2^(k+1)/(x*2^(k+1) + (k+1)/U(k+1) )));(continued fraction, 3rd kind, 4-step ). - Sergei N. Gladkovskii, Dec 01 2012
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MAPLE
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A005126:=-(2-4*z+z**2)/(2*z-1)/(z-1)**2; # Conjectured by Simon Plouffe in his 1992 dissertation
g:=z/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)+n, n=1..34); # Zerinvary Lajos, Jan 11 2009
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MATHEMATICA
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s=2; lst={s}; Do[s+=(s-n); AppendTo[lst, Abs[s]], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)
Table[2^n + n + 1, {n, 0, 30}] (* Wesley Ivan Hurt, Oct 28 2014 *)
LinearRecurrence[{4, -5, 2}, {2, 4, 7}, 40] (* Harvey P. Dale, Aug 18 2016 *)
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PROG
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(MAGMA) [2^n+n+1: n in [0..40]]; // Vincenzo Librandi, Oct 22 2011
(PARI) a(n)=2^n+n+1 \\ Charles R Greathouse IV, Sep 24 2015
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CROSSREFS
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Essentially the same as row sums of A128715.
Cf. A194455.
Sequence in context: A023433 A190168 A288133 * A054151 A018176 A135460
Adjacent sequences: A005123 A005124 A005125 * A005127 A005128 A005129
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KEYWORD
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nonn,easy
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AUTHOR
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Colin Mallows
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EXTENSIONS
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More terms from N. J. A. Sloane, Sep 28 2007
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STATUS
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approved
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