A005126 a(n) = 2^n + n + 1.

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

Offset: 0

Author

Colin Mallows

Keywords

• nonn
• easy

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; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992; arXiv:0911.4975 [math.NT], 2009.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 921
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Essentially the same as row sums of A128715.

Cf. A194455.

Programs

Magma

[2^n+n+1: n in [0..40]]; // Vincenzo Librandi, Oct 22 2011

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 *)

PARI

a(n)=2^n+n+1 \\ Charles R Greathouse IV, Sep 24 2015

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

Extensions

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