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A005408
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The odd numbers: a(n) = 2n+1.
(Formerly M2400)
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443
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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; internal format)
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OFFSET
| 0,2
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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 (amarnath_murthy(AT)yahoo.com), 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 (alexandre.wajnberg(AT)skynet.be), 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 (jhbubby(AT)mindspring.com), Aug 30 2008
Equals INVERT transform of A078050 (signed - Cf. comments); and row sums of triangle A144106. [From Gary W. Adamson, Sep 11 2008]
odd numbers(n) = 2*n+1 = square pyramidal number(3*n+1) / triangular number(3*n+1) [From Pierre CAMI, Sep 27 2008]
A000035(a(n))=1, A059841(a(n))=0. [From Reinhard Zumkeller, Sep 29 2008]
Multiplicative closure of A065091. [From 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. [From Carmine Suriano, Jun 08 2009]
Binomial transform of A130706, inverse binomial transform of A001787(without the initial 0). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]
Also the 3-rough numbers: positive integers that have no prime factors less than 3. [From Michael Porter, Oct 08 2009]
0r n without 2 as prime factor. [From Juri-StepanGerasimov, 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. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Dec 19 2009]
A176271 = odd numbers seen as a triangle read by rows: a(n)=A176271(A002024(n+1),A002260(n+1)). [From 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. [From Jaroslav Krizek, May 28 2010]
Union of A179084 and A179085. [From Reinhard Zumkeller, Jun 28 2010]
For n>0, continued fraction [1,1,n] = (n+1)/a(n); e.g. [1,1,7] = 8/15 [From Gary W. Adamson, Jul 15 2010]
Numbers that are the sum of two sequential integers. [From 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), then ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1=0 (mod h); in our case, a(n)^2-1=0 (mod 4). Also a(n)^2-1=0 (mod 8). [From 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 7 2011]
A001227(a(n)) = A000005(a(n)); A048272(a(n)) < 0. [Reinhard Zumkeller, Jan 21 2012]
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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).
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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
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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
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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).
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 (Hieronymus.Fischer(AT)gmx.de), May 25 2007
a(0)=1, a(1)=3, a(n)=2a(n-1)-a(n-2). - Jaume Oliver i Lafont (joliverlafont(AT)gmail.com), May 07 2008
A005408(n)=A000330(A016777(n))/A000217(A016777(n)) [From Pierre CAMI, Sep 27 2008]
a(n) = A034856(n+1) - A000217(n) = A005843(n) + A000124(n) - A000217(n) = A005843(n) + 1. [From Jaroslav Krizek, Sep 05 2009]
a(n) = (n - 1) + n (sum of two sequential integers) [From Dominick Cancilla, Aug 09 2010]
a(n) = 4*A000217(n)+1 - 2*sum[i=1..n-1] a(i) for n>1. [From 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
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EXAMPLE
| q + 3*q^3 + 5*q^5 + 7*q^7 + 9*q^9 + 11*q^11 + 13*q^13 + 15*q^15 + ...
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MAPLE
| A005408 := n->2*n+1;
A005408:=(1+z)/(z-1)^2; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[2n - 1, {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
Range[1, 131, 2] (* From Harvey P. Dale, Apr 26 2011 *)
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PROG
| (MAGMA) [ 2*n+1 : n in [0..100]];
(PARI) {a(n) = 2*n + 1}
(Haskell)
a005408 n = [1, 3..] !! n -- Reinhard Zumkeller, Jun 28 2011
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CROSSREFS
| Cf. A000027, A005843.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A128200, A000290.
A078050, A144106 [From Gary W. Adamson, Sep 11 2008]
Cf. A109613, A167875. [From Reinhard Zumkeller, Dec 05 2009]
Cf. A001651 (n=1 or 2 mod 3), A047209 (n=1 or 4 mod 5).
Sequence in context: A157142 A004273 * A176271 A144396 A060747 A089684
Adjacent sequences: A005405 A005406 A005407 * A005409 A005410 A005411
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KEYWORD
| easy,nonn,core,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 06 2003
Removed incorrect comment and example Joerg Arndt (arndt(AT)jjj.de), Mar 11 2010
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