login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005408 The odd numbers: a(n) = 2n+1.
(Formerly M2400)
617
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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.

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

Also continued fraction for coth(1) (A073747 is decimal expansion). - Rick L. Shepherd, Aug 07 2002

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

Smallest number greater than n, not a multiple of n, but containing it in binary representation. - Reinhard Zumkeller, Oct 06 2003

Numbers n such that phi(2n)=phi(n), where phi is the Euler's totient(A000010). - Lekraj Beedassy, Aug 27 2004

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

a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - Reinhard Zumkeller, Jun 01 2005

For n>1, numbers having 2 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005

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

First differences of squares (A000290). - Lekraj Beedassy, Jul 15 2006

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

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

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

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

A134451(a(n)) = ABS(A134452(a(n))) = 1; union of A134453 and A134454. - Reinhard Zumkeller, Oct 27 2007

Numbers n such that sigma(2n)=3*sigma(n). - Farideh Firoozbakht, Feb 26 2008

a(n) = A139391(A016825(n)) = A006370(A016825(n)). - Reinhard Zumkeller, Apr 17 2008

Number of divisors of 4^(n-1) for n>0. - J. Lowell, Aug 30 2008

Equals INVERT transform of A078050 (signed - Cf. comments); and row sums of triangle A144106. - Gary W. Adamson, Sep 11 2008

odd numbers(n) = 2*n+1 = square pyramidal number(3*n+1) / triangular number(3*n+1). - Pierre CAMI, Sep 27 2008

A000035(a(n))=1, A059841(a(n))=0. - Reinhard Zumkeller, Sep 29 2008

Multiplicative closure of A065091. - Reinhard Zumkeller, Oct 14 2008

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

Binomial transform of A130706, inverse binomial transform of A001787(without the initial 0). - Philippe Deléham, Sep 17 2009

Also the 3-rough numbers: positive integers that have no prime factors less than 3. - Michael B. Porter, Oct 08 2009

Or n without 2 as prime factor. - Juri-Stepan Gerasimov, Nov 19 2009

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

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

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

Union of A179084 and A179085. - Reinhard Zumkeller, Jun 28 2010

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

Numbers that are the sum of two sequential integers. - Dominick Cancilla, Aug 09 2010

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

A004767 = a(a(n)). - Reinhard Zumkeller, Jun 27 2011

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

A001227(a(n)) = A000005(a(n)); A048272(a(n)) < 0. - Reinhard Zumkeller, Jan 21 2012

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

A007814(a(n)) = 0; A037227(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2012

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

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

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

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

These are also numbers n such that (n^n+1)/(n+1) is an integer. - Derek Orr, May 22 2014

REFERENCES

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

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.

K. Ciesielski and Z. Pogoda, On ordering the natural numbers, or the Sharkovski theorem, Amer. Math. Monthly, 115 (No. 2, 2008), 158-165.

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

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

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

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

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

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 935

Tanya Khovanova, Recursive Sequences

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.

M. Somos, Rational Function Multiplicative Coefficients

William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))

William A. Stein, The modular forms database

Eric Weisstein's World of Mathematics, Odd Number, Davenport-Schinzel Sequence, Gnomonic Number, Pythagorean Triple, Inverse Tangent, Inverse Cotangent, Inverse Hyperbolic Cotangent, Inverse Hyperbolic Tangent, Nexus Number.

Index entries for "core" sequences

Index entries for sequences related to linear recurrences with constant coefficients

FORMULA

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

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

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

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

Euler transform of length 2 sequence [ 3, -1]. - Michael Somos, Mar 30 2007

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

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]

a(n)=(n+1)^2-n^2. G.f. g(x)=sum{k>=0, x^floor(sqr(k))}=sum{k>=0, x^A000196(k)}. - Hieronymus Fischer, May 25 2007

a(0)=1, a(1)=3, a(n)=2a(n-1)-a(n-2). - Jaume Oliver Lafont, May 07 2008

A005408(n)=A000330(A016777(n))/A000217(A016777(n)). - Pierre CAMI, Sep 27 2008

a(n) = A034856(n+1) - A000217(n) = A005843(n) + A000124(n) - A000217(n) = A005843(n) + 1. - Jaroslav Krizek, Sep 05 2009

a(n) = (n - 1) + n (sum of two sequential integers). - Dominick Cancilla, Aug 09 2010

a(n) = 4*A000217(n)+1 - 2*sum[i=1..n-1] a(i) for n>1. - _Bruno Berselli, Nov 17 2010

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

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

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

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

a(n) = floor(sqrt(2*A000384(n+1))). - Ivan N. Ianakiev, Jun 17 2013

a(n) = 3*A000330(n)/A000217(n), n>0. - Ivan N. Ianakiev, Jul 12 2013

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

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

EXAMPLE

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

MAPLE

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

A005408:=(1+z)/(z-1)^2; [Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

Table[2n - 1, {n, 1, 50}] - Stefan Steinerberger, Apr 01 2006

Range[1, 131, 2] (* Harvey P. Dale, Apr 26 2011 *)

PROG

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

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

(Haskell)

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

a005408_list = [1, 3 ..]  -- Reinhard Zumkeller, Feb 11 2012, Jun 28 2011

(Maxima) makelist(2*n+1, n, 0, 30); /* Martin Ettl, Dec 11 2012 */

CROSSREFS

Cf. A000027, A005843, A065091.

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

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

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

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

Cf. A000754 (boustrophedon transform).

Cf. A240876.

Sequence in context: A157142 A004273 * A176271 A144396 A060747 A089684

Adjacent sequences:  A005405 A005406 A005407 * A005409 A005410 A005411

KEYWORD

nonn,easy,core,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Reinhard Zumkeller, Oct 06 2003

Incorrect comment and example removed by Joerg Arndt, Mar 11 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified July 31 15:35 EDT 2014. Contains 245085 sequences.