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A000051 2^n + 1.
(Formerly M0717 N0266)
143
2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Same as Pisot sequence L(2,3).

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

See also A004119 for a(n) = 2a(n-1)-1 with first term = 1. - Philippe Deléham, 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, May 31 2005

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

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

a(n) = A024036(n)/A000225(n). - Reinhard Zumkeller, Feb 14 2009

a(n) = a(n-1)-th odd numbers (A004273) for n >= 1. - Jaroslav Krizek, Apr 25 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)charpoly(A,3). - Milan Janjic, Jan 27 2010

a(n)*A000225(n) = A000225(2*n); a(n) = A173786(n,0). - Reinhard Zumkeller, Feb 28 2010

First differences of A006127. - Reinhard Zumkeller, Apr 14 2011

The odd prime numbers in this sequence form A019434, the Fermat primes. - David W. Wilson, Nov 16 2011

Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4,... - R. J. Mathar, Aug 10 2012

Is the mentioned Pisano period lengths (see above) the same as A007733? - Omar E. Pol, Aug 10 2012

Only positive integers that are not 1 mod (2k+1) for any k>1. - Jon Perry, Oct 16 2012

For n >= 1, a(n) is the total segments length of Hilbert curve after n iterations. - Kival Ngaokrajang, Mar 30 2014

REFERENCES

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).

LINKS

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

Kival Ngaokrajang, Illustration of Hilbert curve for n = 1..5

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Fermat-Lucas Number

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

Eric Weisstein's World of Mathematics, Hilbert curve

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-2)

FORMULA

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

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

First differences of A052944. - Emeric Deutsch, Mar 04 2004

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

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

Equals binomial transform of [2, 1, 1, 1,...]. - Gary W. Adamson, Apr 23 2008

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

E.g.f.: exp(x) + exp(2*x). - Mohammad K. Azarian, Jan 02 2009

From Peter Luschny, 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)

If p[i]=fibonacci(i-4) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010

a(n+2) = a(n) + a(n+1) + A000225(n). - Ivan N. Ianakiev, Jun 24 2012

MAPLE

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

a := n -> add(binomial(n, k)*bernoulli(n-k, 1)*2^(k+1)/(k+1), k=0..n); # Peter Luschny, Apr 20 2009

MATHEMATICA

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

PROG

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

(Haskell)

a000051 = (+ 1) . a000079

a000051_list = iterate ((subtract 1) . (* 2)) 2

-- Reinhard Zumkeller, May 03 2012

CROSSREFS

Apart from the initial 1, identical to A094373.

See A008776 for definitions of Pisot sequences. Cf. A034472, A052539, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081.

Cf. A052944.

Column 2 of array A103438.

Cf. A000079.

Cf. A005126, A176691, A194455.

Sequence in context: A005257 A091697 A109740 * A094373 A213705 A061902

Adjacent sequences:  A000048 A000049 A000050 * A000052 A000053 A000054

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 19 14:44 EDT 2014. Contains 240761 sequences.