0,1

Same as Pisot sequence L(2,3)

Length of the continued fraction for sum(k=0,n,1/3^(2^k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 12 2003

See also A004119 for a(n) = 2a(n-1)-1 with first term =1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004

From the second term on (n>=1), in base 2, these numbers present the pattern 1000...0001 (with n-1 zeros), which is the "opposite" of the binary 2^n-2: (0)111...1110 (cf. A000918). - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), May 31 2005

Numbers n for which the expression 2^n/(n-1) is an integer. - Paolo P. Lava (ppl(AT)spl.at), May 12 2006

a(n) = A127904(n+1) for n>0. - Reinhard Zumkeller, Feb 05 2007

a(n) = A024036(n)/A000225(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 14 2009]

a(n) = a(n-1)-th odd numbers (A004273) for n >= 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Apr 25 2009]

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.

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).

Ivan Panchenko, Table of n, a(n) for n=0..100

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 114

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 362

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence

Index entries for sequences related to linear recurrences with constant coefficients

a(n) = 2a(n-1) - 1 = 3a(n-1) - 2a(n-2).

G.f.: (2-3*x)/((1-x)*(1-2*x)).

First differences of A052944 - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2004

a(0) = 1, then a(n) = (Sum i=0..n-1 a(i)) - (n-2). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jul 10 2004

Inverse binomial transform of A007689. Also, V sequence in Lucas sequence L(3, 2). - Ross La Haye (rlahaye(AT)new.rr.com), Feb 07 2005

Equals binomial transform of [2, 1, 1, 1,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 23 2008

a(n)=A000079(n)+1. - Omar E. Pol (info(AT)polprimos.com), May 18 2008

E.g.f.: e^x+e^(2*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 02 2009]

Contribution from Peter Luschny (peter(AT)luschny.de), Apr 20 2009: (Start)

A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).

a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) (End)

A000051:=-(-2+3*z)/(2*z-1)/(z-1); [S. Plouffe in his 1992 dissertation.]

g:=1/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)+1, n=0..31); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 09 2009]

a := n -> add(binomial(n, k)*bernoulli(n-k, 1)*2^(k+1)/(k+1), k=0..n); [From Peter Luschny (peter(AT)luschny.de), Apr 20 2009]

Table[2^n + 1, {n, 0, 33}]

(PARI) a(n)=if(n<0, 0, 2^n+1)

sage: [lucas_number2(n, 3, 2) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

(Other) sage: [sigma(2, n)for n in xrange(0, 32)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

Apart from the initial 1, identical to A094373..

See A008776 for definitions of Pisot sequences. Cf. A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600 - A074624.

Cf. A052944.

Column 2 of array A103438.

Cf. A000079.

Sequence in context: A005257 A091697 A109740 this_sequence A094373 A061902 A166286

Adjacent sequences: A000048 A000049 A000050 this_sequence A000052 A000053 A000054

easy,nonn

N. J. A. Sloane (njas(AT)research.att.com).