A000051 a(n) = 2^n + 1. (Formerly M0717 N0266)

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

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

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

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 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 - 1)/(n + 1)). This conjecture is inspired by the conjecture of Zhi-Wei Sun in A036968. - Peter Luschny, Sep 07 2021

References

Paul 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

E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]

Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.

Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).

Bartomeu Fiol, Jairo Martínez-Montoya, and Alan Rios Fukelman, The planar limit of N=2 superconformal field theories, arXiv:2003.02879 [hep-th], 2020.

Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 114

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 362

Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

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; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).

Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.

Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].

Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.

Eric Weisstein's World of Mathematics, Crossed Prism Graph.

Eric Weisstein's World of Mathematics, Cunningham Number.

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

Eric Weisstein's World of Mathematics, Hilbert curve.

Eric Weisstein's World of Mathematics, Independent Vertex Set.

Eric Weisstein's World of Mathematics, Irredundant Set.

Eric Weisstein's World of Mathematics, Matching Number.

Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.

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

Eric Weisstein's World of Mathematics, Star Graph.

Eric Weisstein's World of Mathematics, Vertex Cover.

Index entries for 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

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

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

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

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

From Reinhard Zumkeller, Feb 28 2010: (Start)

a(n)*A000225(n) = A000225(2*n).

a(n) = A173786(n,0). (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

a(A006521(n)) mod A006521(n) = 0. - Reinhard Zumkeller, Jul 17 2014

a(n) = 3*A007583((n-1)/2) for n odd. - Eric W. Weisstein, Jul 17 2017

Sum_{n>=0} 1/a(n) = A323482. - Amiram Eldar, Nov 11 2020

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}]
2^Range[0, 20] + 1 (* Eric W. Weisstein, Jul 17 2017 *)
LinearRecurrence[{3, -2}, {2, 3}, 20] (* 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
-- Reinhard Zumkeller, May 03 2012
(Python)
def A000051(n): return (1<<n)|1 if n else 2 # Chai Wah Wu, Dec 21 2022

Crossrefs

Apart from the initial 1, identical to A094373.

See A008776 for definitions of Pisot sequences.

Cf. A034472, A052539, A034474, A062394, A034491, A036968, A062395, A062396, A062397, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081, A173786, A052944, A000079, A005126, A176691, A194455, A323482.

Column 2 of array A103438.

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

Sequence in context: A091697 A109740 A248155 * A094373 A213705 A295637

Adjacent sequences: A000048 A000049 A000050 * A000052 A000053 A000054

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved