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A002064
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Cullen numbers: n*2^n + 1.
(Formerly M2795 N1125)
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37
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1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry (pbarry(AT)wit.ie), Jun 12 2003
Equals row sums of triangle A143038 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 18 2008
Equals row sums of triangle A156708 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 13 2009]
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). [From Milan R. Janjic (agnus(AT)blic.net), Jan 26 2010]
Grau proved that that there is no Cullen number with the Lehmer property. Hence, if phi (C_n) | C_n - 1, then C_n is prime. A composite integer m is called a Lehmer number if phi(m) | m - 1, where phi(m) is the Euler function of m = A000010(m). - Jonathan Vos Post, Mar 20 2011
A181527 = Partial sums of A002064 -- Vladimir Joseph Stephan Orlovsky, Jul 09 2011.
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REFERENCES
| G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
R. K. Guy, Unsolved Problems in Number Theory, B20.
W. Sierpi\'{n}ski, Elementary Theory of Numbers. Pa\'{n}st. Wydaw. Nauk., Warsaw, 1964, p. 346.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..300
Index entries for sequences related to linear recurrences with constant coefficients
Ray Ballinger, Cullen Primes: Definition and Status
C. K. Caldwell, Cullen Primes
Jose Maria Grau, Florian Luca, Cullen Numbers with the Lehmer Property, Mar 18, 2011.
Paul Leyland, Factors of Cullen and Woodall numbers
Paul Leyland, Generalized Cullen and Woodall numbers
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
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.
W. Sierpi\'{n}ski, Elementary Theory of Numbers, Warszawa 1964.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wikipedia, Cullen number
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FORMULA
| a(n)=4a(n-1)-4a(n-2)+1. - Paul Barry (pbarry(AT)wit.ie), Jun 12 2003
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
Row sums of triangle A134081. Equals A001787(n) - (2^n - 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 07 2007
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
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
a(0)=1, a(1)=3, a(2)=9, a(n)=5*a(n-1)-8*a(n-2)+4*a(n-3) [From Harvey P. Dale, Oct 13 2011]
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MAPLE
| A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[n*2^n+1, {n, 0, 2*4!}] (* From Vladimir Orlovsky, Apr 25 2010 *)
LinearRecurrence[{5, -8, 4}, {1, 3, 9}, 51] (* From Harvey P. Dale, Oct 13 2011 *)
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CROSSREFS
| Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.
Sequence in context: A145127 A096260 A195417 * A129589 A096322 A058396
Adjacent sequences: A002061 A002062 A002063 * A002065 A002066 A002067
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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