Search: id:A002064
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%I A002064 M2795 N1125
%S A002064 1,3,9,25,65,161,385,897,2049,4609,10241,22529,49153,106497,229377,
%T A002064 491521,1048577,2228225,4718593,9961473,20971521,44040193,92274689,
%U A002064 192937985,402653185,838860801,1744830465,3623878657,7516192769
%N A002064 Cullen numbers: n*2^n + 1.
%C A002064 Binomial transform is A084859. Inverse binomial transform is A004277. 
               - Paul Barry (pbarry(AT)wit.ie), Jun 12 2003
%C A002064 Equals row sums of triangle A143038 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jul 18 2008
%C A002064 Equals row sums of triangle A156708 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Feb 13 2009]
%D A002064 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002064 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002064 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A002064 R. K. Guy, Unsolved Problems in Number Theory, B20.
%D A002064 W. Sierpi\'{n}ski, Elementary Theory of Numbers. Pa\'{n}st. Wydaw. Nauk., 
               Warsaw, 1964, p. 346.
%H A002064 T. D. Noe, <a href="b002064.txt">Table of n, a(n) for n=0..300</a>
%H A002064 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002064 Ray Ballinger, <a href="http://www.prothsearch.net/cullen.html">Cullen 
               Primes: Definition and Status</a>
%H A002064 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/lists/top20/
               Cullen.html">Cullen Primes</a>
%H A002064 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/
               cullen_woodall/cw.htm">Factors of Cullen and Woodall numbers</a>
%H A002064 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/factorization/
               cullen_woodall/gcw.htm">Generalized Cullen and Woodall numbers</a>
%H A002064 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha103.htm">Factorizations of many number sequences</a>
%H A002064 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha118.htm">Factorizations of many number sequences</a>
%H A002064 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha119.htm">Factorizations of many number sequences</a>
%H A002064 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha120.htm">Factorizations of many number sequences</a>
%H A002064 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha121.htm">Factorizations of many number sequences</a>
%H A002064 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 A002064 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 A002064 W. Sierpi\'{n}ski, <a href="http://matwbn.icm.edu.pl/kstresc.php?tom=42&wyd=10">
               Elementary Theory of Numbers</a>, Warszawa 1964.
%H A002064 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CullenNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A002064 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cullen_prime">Cullen 
               number</a>
%F A002064 a(n)=4a(n-1)-4a(n-2)+1. - Paul Barry (pbarry(AT)wit.ie), Jun 12 2003
%F A002064 a(n) = sum of row (n+1) of triangle A130197. Example: a(3) = 25 = (12 
               + 8 + 4 + 1), row 4 of A130197. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 16 2007
%F A002064 Row sums of triangle A134081. Equals A001787(n) - (2^n - 1). - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Oct 07 2007
%F A002064 G.f.: -(1-2*x+2*x^2)/((-1+x)*(2*x-1)^2). a(n)=A001787(n+1)+1-A000079(n). 
               - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
%F A002064 a(n) = 1 + 2^(n + log_2(n)) ~ 1 + A000079(n+A004257(n)). a(n) ~ A000051(n+A004257(n)). 
               - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 20 2008
%p A002064 A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); [Conjectured by S. Plouffe 
               in his 1992 dissertation.]
%Y A002064 Cf. A005849, A003261, A050914, A130197, A134081, A001787.
%Y A002064 Cf. A143038.
%Y A002064 A156708 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 13 2009]
%Y A002064 Sequence in context: A065971 A145127 A096260 this_sequence A129589 A096322 
               A058396
%Y A002064 Adjacent sequences: A002061 A002062 A002063 this_sequence A002065 A002066 
               A002067
%K A002064 nonn,easy,nice
%O A002064 0,2
%A A002064 N. J. A. Sloane (njas(AT)research.att.com).

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