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A005126 2^n + n + 1.
(Formerly M1061)
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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 and Conjectures, Université du Québec à Montréal, 1992.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 921

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Colin Mallows

EXTENSIONS

More terms from N. J. A. Sloane, Sep 28 2007

STATUS

approved

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Last modified June 23 21:29 EDT 2017. Contains 288675 sequences.