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Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A005126 a(n) = 2^n + n + 1. (Formerly M1061) 25 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; 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). 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 A374729 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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