A000051 - OEIS

%I M0717 N0266 #241 Feb 07 2024 01:12:49

%S 2,3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,

%T 131073,262145,524289,1048577,2097153,4194305,8388609,16777217,

%U 33554433,67108865,134217729,268435457,536870913,1073741825,2147483649,4294967297,8589934593

%N a(n) = 2^n + 1.

%C Same as Pisot sequence L(2,3).

%C Length of the continued fraction for Sum_{k=0..n} 1/3^(2^k). - _Benoit Cloitre_, Nov 12 2003

%C See also A004119 for a(n) = 2a(n-1)-1 with first term = 1. - _Philippe Deléham_, Feb 20 2004

%C 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

%C 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

%C First differences of A006127. - _Reinhard Zumkeller_, Apr 14 2011

%C The odd prime numbers in this sequence form A019434, the Fermat primes. - _David W. Wilson_, Nov 16 2011

%C 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

%C Is the mentioned Pisano period lengths (see above) the same as A007733? - _Omar E. Pol_, Aug 10 2012

%C Only positive integers that are not 1 mod (2k+1) for any k>1. - _Jon Perry_, Oct 16 2012

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

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

%C 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

%C 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

%C 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

%C Also the number of maximum matchings in the 2(n-1)-crossed prism graph. - _Eric W. Weisstein_, Dec 31 2017

%C 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

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ivan Panchenko, <a href="/A000051/b000051.txt">Table of n, a(n) for n = 0..100</a>

%H E. R. Berlekamp, <a href="/A257113/a257113.pdf">A contribution to mathematical psychometrics</a>, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]

%H Bakir Farhi, <a href="https://www.emis.de/journals/JIS/VOL22/Farhi/farhi19.html">Summation of Certain Infinite Lucas-Related Series</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.

%H Massimiliano Fasi and Gian Maria Negri Porzio, <a href="http://eprints.maths.manchester.ac.uk/2709/">Determinants of Normalized Bohemian Upper Hessemberg Matrices</a>, University of Manchester (England, 2019).

%H Bartomeu Fiol, Jairo Martínez-Montoya, and Alan Rios Fukelman, <a href="https://arxiv.org/abs/2003.02879">The planar limit of N=2 superconformal field theories</a>, arXiv:2003.02879 [hep-th], 2020.

%H Bernard Frénicle de Bessy, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k3187867/f7.item">Solutio duorum problematum circa numeros cubos et quadratos</a>, (1657). Bibliothèque Nationale de Paris.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=114">Encyclopedia of Combinatorial Structures 114</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=362">Encyclopedia of Combinatorial Structures 362</a>

%H Edouard Lucas, <a href="http://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf">The Theory of Simply Periodic Numerical Functions</a>, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

%H Kival Ngaokrajang, <a href="/A000051/a000051_1.pdf">Illustration of Hilbert curve for n = 1..5</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.2634312">On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers</a>, Politecnico di Torino (Italy, 2019).

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2044">Composition Operations of Generalized Entropies Applied to the Study of Numbers</a>, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences (2019) Vol. 8, No. 10.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrossedPrismGraph.html">Crossed Prism Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CunninghamNumber.html">Cunningham Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fermat-LucasNumber.html">Fermat-Lucas Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HilbertCurve.html">Hilbert curve</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrredundantSet.html">Irredundant Set</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumIndependentEdgeSet.html">Maximum Independent Edge Set</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rudin-ShapiroSequence.html">Rudin-Shapiro Sequence</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarGraph.html">Star Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).

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

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

%F First differences of A052944. - _Emeric Deutsch_, Mar 04 2004

%F a(0) = 1, then a(n) = (Sum_{i=0..n-1} a(i)) - (n-2). - _Gerald McGarvey_, Jul 10 2004

%F Inverse binomial transform of A007689. Also, V sequence in Lucas sequence L(3, 2). - _Ross La Haye_, Feb 07 2005

%F a(n) = A127904(n+1) for n>0. - _Reinhard Zumkeller_, Feb 05 2007

%F Equals binomial transform of [2, 1, 1, 1, ...]. - _Gary W. Adamson_, Apr 23 2008

%F a(n) = A000079(n)+1. - _Omar E. Pol_, May 18 2008

%F E.g.f.: exp(x) + exp(2*x). - _Mohammad K. Azarian_, Jan 02 2009

%F a(n) = A024036(n)/A000225(n). - _Reinhard Zumkeller_, Feb 14 2009

%F From _Peter Luschny_, Apr 20 2009: (Start)

%F A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).

%F a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) (End)

%F a(n) is the a(n-1)-th odd number for n >= 1. - _Jaroslav Krizek_, Apr 25 2009

%F From _Reinhard Zumkeller_, Feb 28 2010: (Start)

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

%F a(n) = A173786(n,0). (End)

%F 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

%F a(n+2) = a(n) + a(n+1) + A000225(n). - _Ivan N. Ianakiev_, Jun 24 2012

%F a(A006521(n)) mod A006521(n) = 0. - _Reinhard Zumkeller_, Jul 17 2014

%F a(n) = 3*A007583((n-1)/2) for n odd. - _Eric W. Weisstein_, Jul 17 2017

%F Sum_{n>=0} 1/a(n) = A323482. - _Amiram Eldar_, Nov 11 2020

%p A000051:=-(-2+3*z)/(2*z-1)/(z-1); # _Simon Plouffe_ in his 1992 dissertation

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

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

%t 2^Range[0, 20] + 1 (* _Eric W. Weisstein_, Jul 17 2017 *)

%t LinearRecurrence[{3, -2}, {2, 3}, 20] (* _Eric W. Weisstein_, Sep 21 2017 *)

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

%o (PARI) first(n) = Vec((2 - 3*x)/((1 - x)*(1 - 2*x)) + O(x^n)) \\ _Iain Fox_, Dec 31 2017

%o (Haskell)

%o a000051 = (+ 1) . a000079

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

%o -- _Reinhard Zumkeller_, May 03 2012

%o (Python)

%o def A000051(n): return (1<<n)|1 if n else 2 # _Chai Wah Wu_, Dec 21 2022

%Y Apart from the initial 1, identical to A094373.

%Y See A008776 for definitions of Pisot sequences.

%Y 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.

%Y Column 2 of array A103438.

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

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_