A000051 - OEIS (original) (raw)

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, 4294967297, 8589934593

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

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

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

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 length of the segments of the Hilbert curve after n iterations. - Kival Ngaokrajang, Mar 30 2014

Frénicle de Bessy (1657) proved that a(3) = 9 is the only square in this sequence. - Charles R Greathouse IV, May 13 2014

a(n) is the number of distinct possible sums made with at most two elements in {1,...,a(n-1)} for n > 0. - Derek Orr, Dec 13 2014

For n > 0, given any set of a(n) lattice points in R^n, there exist 2 distinct members in this set whose midpoint is also a lattice point. - Melvin Peralta, Jan 28 2017

Also the number of independent vertex sets, irredundant sets, and vertex covers in the (n+1)-star graph. - Eric W. Weisstein, Aug 04 and Sep 21 2017

Also the number of maximum matchings in the 2(n-1)-crossed prism graph. - Eric W. Weisstein, Dec 31 2017

Conjecture: For any integer n >= 0, a(n) is the permanent of the (n+1) X (n+1) matrix with M(j, k) = -floor((j - k - 2)/(n + 1)). This conjecture is inspired by the conjecture of Zhi-Wei Sun in A036968. - Peter Luschny, Sep 07 2021

The corrected Luschny's conjecture is true (see Fried link). - Sela Fried, Dec 02 2025

Also the number of maximum matchings and minimum edge covers in the n-necklace graph. - Eric W. Weisstein, Feb 12-13 2026

REFERENCES

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

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 46, 60, 244.

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).

Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, exercise 5.34 on pages 174-175.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 141.

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)).

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 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)

a(n) is the a(n-1)-th odd number for n >= 1. - Jaroslav Krizek, Apr 25 2009

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

MAPLE

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, 40}]

LinearRecurrence[{3, -2}, {2, 3}, 40] (* Eric W. Weisstein, Sep 21 2017 *)

PROG

(PARI) a(n)=2^n+1

(PARI) first(n) = Vec((2 - 3*x)/((1 - x)*(1 - 2*x)) + O(x^n)) \\ Iain Fox, Dec 31 2017

(Haskell)

a000051 = (+ 1) . a000079

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

(Python)

(Magma) [2^n+1: n in [0..40]]; // G. C. Greubel, Jan 18 2025

CROSSREFS

Apart from the initial 1, identical to A094373.

See A008776 for definitions of Pisot sequences.

Cf. A007583 (a((n-1)/2)/3 for odd n).