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A005408 The odd numbers: a(n) = 2n+1.
(Formerly M2400)
659

%I M2400

%S 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,

%T 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,

%U 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131

%N The odd numbers: a(n) = 2n+1.

%C Leibniz's series: Pi/4 = Sum_{n=0..inf} (-1)^n/(2n+1) (Cf. A072172).

%C Beginning of the ordering of the natural numbers used in Sharkovski's theorem - see the Cielsielski-Pogoda paper.

%C The Sharkovski ordering begins with the odd numbers >= 3, then twice these numbers, then 4 times them, then 8 times them, etc., ending with the powers of 2 in decreasing order, ending with 2^0 = 1.

%C Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 6 ).

%C Also continued fraction for coth(1) (A073747 is decimal expansion). - _Rick L. Shepherd_, Aug 07 2002

%C a(1) = 1; a(n) = smallest number such that a(n) + a(i) is composite for all i = 1 to n-1. - _Amarnath Murthy_, Jul 14 2003

%C Smallest number greater than n, not a multiple of n, but containing it in binary representation. - _Reinhard Zumkeller_, Oct 06 2003

%C Numbers n such that phi(2n)=phi(n), where phi is the Euler's totient (A000010). - _Lekraj Beedassy_, Aug 27 2004

%C Pi*sqrt(2)/4 = Sum_{n=0..inf} (-1)^floor(n/2)/(2n+1) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 ... [since periodic f(x)=x over -Pi<x<Pi = 2(sin(x)/1 - sin(2x)/2 + sin(3x)/3 - ...) using x = Pi/4 (Maor)]. - _Gerald McGarvey_, Feb 04 2005

%C a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - _Reinhard Zumkeller_, Jun 01 2005

%C For n>1, numbers having 2 as an anti-divisor. - _Alexandre Wajnberg_, Oct 02 2005

%C a(n) = shortest side a of all integer-sided triangles with sides a<=b<=c and inradius n >= 1.

%C First differences of squares (A000290). - _Lekraj Beedassy_, Jul 15 2006

%C The odd numbers are the solution to the simplest recursion arising when assuming that the algorithm "merge sort" could merge in constant unit time, i.e., T(1):= 1, T(n):= T(floor(n/2)) + T(ceiling(n/2)) + 1. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 14 2006

%C 2n-5 counts the permutations in S_n which have zero occurrences of the pattern 312 and one occurrence of the pattern 123. - David Hoek (david.hok(AT)telia.com), Feb 28 2007

%C For n>0: number of divisors of (n-1)th power of any squarefree semiprime: a(n)=A000005(A001248(k)^(n-1)); a(n) = A000005(A000302(n-1)) = A000005(A001019(n-1)) = A000005(A009969(n-1)) = A000005(A087752(n-1)). - _Reinhard Zumkeller_, Mar 04 2007

%C For n > 2, a(n-1) is the least integer not the sum of < n n-gonal numbers (0 allowed). - _Jonathan Sondow_, Jul 01 2007

%C A134451(a(n)) = ABS(A134452(a(n))) = 1; union of A134453 and A134454. - _Reinhard Zumkeller_, Oct 27 2007

%C Numbers n such that sigma(2n) = 3*sigma(n). - _Farideh Firoozbakht_, Feb 26 2008

%C a(n) = A139391(A016825(n)) = A006370(A016825(n)). - _Reinhard Zumkeller_, Apr 17 2008

%C Number of divisors of 4^(n-1) for n > 0. - _J. Lowell_, Aug 30 2008

%C Equals INVERT transform of A078050 (signed - Cf. comments); and row sums of triangle A144106. - _Gary W. Adamson_, Sep 11 2008

%C odd numbers(n) = 2*n+1 = square pyramidal number(3*n+1) / triangular number(3*n+1). - _Pierre CAMI_, Sep 27 2008

%C A000035(a(n))=1, A059841(a(n))=0. - _Reinhard Zumkeller_, Sep 29 2008

%C Multiplicative closure of A065091. - _Reinhard Zumkeller_, Oct 14 2008

%C a(n) is also the maximum number of triangles that n+2 points in the same plane can determine. 3 points determine max 1 triangle; 4 points can give 3 triangles; 5 points can give 5; 6 points can give 7 etc. - _Carmine Suriano_, Jun 08 2009

%C Binomial transform of A130706, inverse binomial transform of A001787(without the initial 0). - _Philippe Deléham_, Sep 17 2009

%C Also the 3-rough numbers: positive integers that have no prime factors less than 3. - _Michael B. Porter_, Oct 08 2009

%C Or n without 2 as prime factor. - _Juri-Stepan Gerasimov_, Nov 19 2009

%C Given an L(2,1) labeling l of a graph G, let k be the maximum label assigned by l. The minimum k possible over all L(2,1) labelings of G is denoted by lambda(G). For n > 0, this sequence gives lambda(K_{n+1}) where K_{n+1} is the complete graph on n+1 vertices. - _K.V.Iyer_, Dec 19 2009

%C A176271 = odd numbers seen as a triangle read by rows: a(n)=A176271(A002024(n+1),A002260(n+1)). - _Reinhard Zumkeller_, Apr 13 2010

%C For n >= 1, a(n-1) = numbers k such that arithmetic mean of the first k positive integers is integer. A040001(a(n-1)) = 1. See A145051 and A040001. - _Jaroslav Krizek_, May 28 2010

%C Union of A179084 and A179085. - _Reinhard Zumkeller_, Jun 28 2010

%C For n>0, continued fraction [1,1,n] = (n+1)/a(n); e.g., [1,1,7] = 8/15. - _Gary W. Adamson_, Jul 15 2010

%C Numbers that are the sum of two sequential integers. - _Dominick Cancilla_, Aug 09 2010

%C Cf. property described by _Gary Detlefs_ in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h and n in A000027), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 4). Also a(n)^2-1=0 (mod 8). - _Bruno Berselli_, Nov 17 2010

%C A004767 = a(a(n)). - _Reinhard Zumkeller_, Jun 27 2011

%C For n>=3 they are the numbers for which the product of their proper divisors divides the product of their anti-divisors. - _Paolo P. Lava_, Jul 07 2011

%C A001227(a(n)) = A000005(a(n)); A048272(a(n)) < 0. - _Reinhard Zumkeller_, Jan 21 2012

%C a(n) is the minimum number of tosses of a fair coin needed so that the probability of more than n heads is at least 1/2. In fact, sum(Pr(k heads|2n+1 tosses),k=n+1,...,2n+1) = 1/2. - _Dennis P. Walsh_, Apr 04 2012

%C A007814(a(n)) = 0; A037227(a(n)) = 1. - _Reinhard Zumkeller_, Jun 30 2012

%C 1/N, (i.e. 1/1, 1/2, 1/3,...) = Sum_{j=1,3,5,..inf} k^j, where k is the infinite set of constants 1/(exp.ArcSinh(N/2) = convergents to barover[N]. The convergent to barover[1} or [1,1,1,...] = 1/Phi = .6180339...; whereas c.f. barover[2] converges to .414213...; and so on. Thus, with k = 1/Phi we obtain 1 = k^1 + k^3 + k^5 + ...; and with k = .414213...= (sqrt(2) - 1) we get 1/2 = k^1 + k^3 + k^5 + ..... Likewise with the convergent to barover[3] = .302775... = k, we get 1/3 = k^1 + k^3 + k^5 + ...; etc. - _Gary W. Adamson_, Jul 01 2012

%C Conjecture on primes with one coach (A216371) relating to the odd integers: iff an integer is in A216371 (primes with one coach either of the form 4q-1 or 4q+1, (q>0)); the top row of its coach is composed of a permutation of the first q odd integers. Example: prime 19 (q = 5), has 5 terms in each row of its coach: 19: [1, 9, 5, 7, 3] ... [1, 1, 1, 2, 4]. This is interpreted: (19 - 1) = (2^1 * 9), (19 - 9) = (2^1 * 5), (19 - 5) = (2^1 - 7), (19 - 7) = (2^2 * 3), (19 - 3) = (2^4 * 1). - _Gary W. Adamson_, Sep 09 2012

%C A005408 is the numerator 2n-1 of the term (1/m^2-1/n^2)=(2n-1)/(mn)^2,n=m+1,m>0 in the Rydberg formula, while A035287 is the denominator (mn)^2. So the quotient a(A005408)/a(A035287) simulates the Hydrogen spectral series of all hydrogen-like elements. - _Freimut Marschner_, Aug 10 2013

%C This sequence has unique factorization. The primitive elements are the odd primes (A065091). (Each term of the sequence can be expressed as a product of terms of the sequence. Primitive elements have only the trivial factorization. If the products of terms of the sequence are always in the sequence, and there is a unique factorization of each element into primitive elements, we say that the sequence has unique factorization. So, e.g., the composite numbers do not have unique factorization, because for example 36 = 4*9 = 6*6 has two distinct factorizations.) - _Franklin T. Adams-Watters_, Sep 28 2013

%C These are also numbers n such that (n^n+1)/(n+1) is an integer. - _Derek Orr_, May 22 2014

%C a(n-1) gives the number of distinct sums in the direct sum {1,2,3,..,n} + {1,2,3,..,n}. For example, {1} + {1} has only one possible sum so a(0) = 1. {1,2} + {1,2} has three distinct possible sums {2,3,4} so a(1) = 3. {1,2,3} + {1,2,3} has 5 distinct possible sums {2,3,4,5,6} so a(2) = 5. - _Derek Orr_, Nov 22 2014

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.

%D T. Dantzig, The Language of Science, 4th Edition (1954) page 276.

%D H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73.

%D D. Hoek, Parvisa moenster i permutationer [Swedish], (2007).

%D E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 203-205.

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

%H N. J. A. Sloane, <a href="/A005408/b005408.txt">Table of n, a(n) for n = 0..10000</a>

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.

%H Hongwei Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Chen/chen78.html">Evaluations of Some Variant Euler Sums</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.

%H K. Ciesielski and Z. Pogoda, <a href="http://www.jstor.org/stable/27642424">On ordering the natural numbers, or the Sharkovski theorem</a>, Amer. Math. Monthly, 115 (No. 2, 2008), 158-165.

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=935">Encyclopedia of Combinatorial Structures 935</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>

%H William A. Stein, <a href="http://modular.math.washington.edu/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>

%H William A. Stein, <a href="http://modular.math.washington.edu/Tables/">The modular forms database</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddNumber.html">Odd Number</a>, <a href="http://mathworld.wolfram.com/Davenport-SchinzelSequence.html">Davenport-Schinzel Sequence</a>, <a href="http://mathworld.wolfram.com/GnomonicNumber.html">Gnomonic Number</a>, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>, <a href="http://mathworld.wolfram.com/InverseTangent.html">Inverse Tangent</a>, <a href="http://mathworld.wolfram.com/InverseCotangent.html">Inverse Cotangent</a>, <a href="http://mathworld.wolfram.com/InverseHyperbolicCotangent.html">Inverse Hyperbolic Cotangent</a>, <a href="http://mathworld.wolfram.com/InverseHyperbolicTangent.html">Inverse Hyperbolic Tangent</a>, <a href="http://mathworld.wolfram.com/NexusNumber.html">Nexus Number</a>.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Rec#order_02">Index to sequences with linear recurrences with constant coefficients</a>, signature (2,-1).

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

%F G.f.: (1 + x) / (1 - x)^2.

%F E.g.f.: (1 + 2*x) * exp(x).

%F G.f. with interpolated zeros: (x^3+x)/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*(exp(x)+exp(-x))/2. - _Geoffrey Critzer_, Aug 25 2012

%F Euler transform of length 2 sequence [ 3, -1]. - _Michael Somos_, Mar 30 2007

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 2*u) * (1 - 2*u + 16*v) - (u - 4*v)^2 * (1 + 2*u + 2*u^2). - _Michael Somos_, Mar 30 2007

%F a(n) = b(2*n + 1) where b(n) = n if n is odd is multiplicative. [This seems to say that A000027 is multiplicative? - _R. J. Mathar_, Sep 23 2011]

%F a(n) = (n+1)^2-n^2. G.f. g(x) = Sum_{k=0..inf} x^floor(sqrt(k)) = Sum_{k=0..inf} x^A000196(k). - _Hieronymus Fischer_, May 25 2007

%F a(0) = 1, a(1) = 3, a(n) = 2*a(n-1)-a(n-2). - _Jaume Oliver Lafont_, May 07 2008

%F A005408(n)=A000330(A016777(n))/A000217(A016777(n)). - _Pierre CAMI_, Sep 27 2008

%F a(n) = A034856(n+1) - A000217(n) = A005843(n) + A000124(n) - A000217(n) = A005843(n) + 1. - _Jaroslav Krizek_, Sep 05 2009

%F a(n) = (n - 1) + n (sum of two sequential integers). - _Dominick Cancilla_, Aug 09 2010

%F a(n) = 4*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. - _Bruno Berselli_, Nov 17 2010

%F n*a(2n+1)^2+1 = (n+1)*a(2n)^2; e.g., 3*15^2+1 = 4*13^2. - _Charlie Marion_, Dec 31 2010

%F arctanh(x) = sum_{n=0..infinity} x^(2n+1)/a(n). - _R. J. Mathar_, Sep 23 2011

%F a(n) = det(f(i-j+1))_{1<=i,j<=n}, where f(n) = A113311(n); for n<0 we have f(n)=0. - _Mircea Merca_, Jun 23 2012

%F G.f.: Q(0), where Q(k)= 1 + 2*(k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 11 2013

%F a(n) = floor(sqrt(2*A000384(n+1))). - _Ivan N. Ianakiev_, Jun 17 2013

%F a(n) = 3*A000330(n)/A000217(n), n>0. - _Ivan N. Ianakiev_, Jul 12 2013

%F a(n) = product(2*sin(Pi*k/(2*n+1)), k=1..2*n) = product((2*sin(Pi*k/(2*n+1)))^2, k=1..n), n>=0 (undefined product = 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - _Wolfdieter Lang_, Oct 10 2013

%F Noting that as n -> infinity, sqrt(n^2 + n) -> n + 1/2, let f(n) = n + 1/2 - sqrt(n^2 + n). Then for n > 0, a(n) = round(1/f(n))/4. - _Richard R. Forberg_, Feb 16 2014

%e q + 3*q^3 + 5*q^5 + 7*q^7 + 9*q^9 + 11*q^11 + 13*q^13 + 15*q^15 + ...

%p A005408 := n->2*n+1;

%p A005408:=(1+z)/(z-1)^2; [_Simon Plouffe_ in his 1992 dissertation.]

%t Table[2n - 1, {n, 1, 50}] - _Stefan Steinerberger_, Apr 01 2006

%t Range[1,131,2] (* _Harvey P. Dale_, Apr 26 2011 *)

%o (MAGMA) [ 2*n+1 : n in [0..100]];

%o (PARI) {a(n) = 2*n + 1}

%o (Haskell)

%o a005408 n = (+ 1) . (* 2)

%o a005408_list = [1, 3 ..] -- _Reinhard Zumkeller_, Feb 11 2012, Jun 28 2011

%o (Maxima) makelist(2*n+1, n, 0, 30); /* _Martin Ettl_, Dec 11 2012 */

%Y Cf. A000027, A005843, A065091.

%Y See A120062 for sequences related to integer-sided triangles with integer inradius n.

%Y Cf. A128200, A000290, A078050, A144106, A109613, A167875.

%Y Cf. A001651 (n=1 or 2 mod 3), A047209 (n=1 or 4 mod 5).

%Y Cf. A003558, A216371, A179480 (relating to the Coach theorem).

%Y Cf. A000754 (boustrophedon transform).

%Y Cf. A240876.

%K nonn,easy,core,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Reinhard Zumkeller_, Oct 06 2003

%E Incorrect comment and example removed by _Joerg Arndt_, Mar 11 2010

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Last modified December 20 23:03 EST 2014. Contains 252290 sequences.