Search: id:A000051
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%I A000051 M0717 N0266
%S A000051 2,3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,
%T A000051 131073,262145,524289,1048577,2097153,4194305,8388609,16777217,
%U A000051 33554433,67108865,134217729,268435457,536870913,1073741825,2147483649
%N A000051 2^n + 1.
%C A000051 Same as Pisot sequence L(2,3)
%C A000051 Length of the continued fraction for sum(k=0,n,1/3^(2^k)) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Nov 12 2003
%C A000051 See also A004119 for a(n) = 2a(n-1)-1 with first term =1 . - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004
%C A000051 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
%C A000051 Numbers n for which the expression 2^n/(n-1) is an integer. - Paolo P. 
               Lava (ppl(AT)spl.at), May 12 2006
%C A000051 a(n) = A127904(n+1) for n>0. - Reinhard Zumkeller, Feb 05 2007
%C A000051 a(n) = A024036(n)/A000225(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 14 2009]
%C A000051 a(n) = a(n-1)-th odd numbers (A004273) for n >= 1. [From Jaroslav Krizek 
               (jaroslav.krizek(AT)atlas.cz), Apr 25 2009]
%D A000051 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 
               1968, vol. 2, p. 75.
%D A000051 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000051 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000051 Ivan Panchenko, <a href="b000051.txt">Table of n, a(n) for n=0..100</
               a>
%H A000051 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=114">
               Encyclopedia of Combinatorial Structures 114</a>
%H A000051 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=362">
               Encyclopedia of Combinatorial Structures 362</a>
%H A000051 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000051 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000051 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Fermat-LucasNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000051 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rudin-ShapiroSequence.html">Rudin-Shapiro Sequence</a>
%H A000051 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000051 a(n) = 2a(n-1) - 1 = 3a(n-1) - 2a(n-2).
%F A000051 G.f.: (2-3*x)/((1-x)*(1-2*x)).
%F A000051 First differences of A052944 - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Mar 04 2004
%F A000051 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
%F A000051 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
%F A000051 Equals binomial transform of [2, 1, 1, 1,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Apr 23 2008
%F A000051 a(n)=A000079(n)+1. - Omar E. Pol (info(AT)polprimos.com), May 18 2008
%F A000051 E.g.f.: e^x+e^(2*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Jan 02 2009]
%F A000051 Contribution from Peter Luschny (peter(AT)luschny.de), Apr 20 2009: (Start)
%F A000051 A weighted binomial sum of the Bernoulli numbers A027641/A027642 with 
               A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).
%F A000051 a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) 
               (End)
%p A000051 A000051:=-(-2+3*z)/(2*z-1)/(z-1); [S. Plouffe in his 1992 dissertation.]
%p A000051 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]
%p A000051 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]
%t A000051 Table[2^n + 1, {n, 0, 33}]
%o A000051 (PARI) a(n)=if(n<0,0,2^n+1)
%o A000051 sage: [lucas_number2(n,3,2) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 25 2008
%o A000051 (Other) sage: [sigma(2,n)for n in xrange(0,32)] # [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 04 2009]
%Y A000051 Apart from the initial 1, identical to A094373..
%Y A000051 See A008776 for definitions of Pisot sequences. Cf. A034472, A052539, 
               A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, 
               A074600 - A074624.
%Y A000051 Cf. A052944.
%Y A000051 Column 2 of array A103438.
%Y A000051 Cf. A000079.
%Y A000051 Sequence in context: A005257 A091697 A109740 this_sequence A094373 A061902 
               A166286
%Y A000051 Adjacent sequences: A000048 A000049 A000050 this_sequence A000052 A000053 
               A000054
%K A000051 easy,nonn
%O A000051 0,1
%A A000051 N. J. A. Sloane (njas(AT)research.att.com).

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