%I A005408 M2400
%S A005408 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,
%T A005408 51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,
%U A005408 97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131
%N A005408 The odd numbers: a(n) = 2n+1.
%C A005408 Leibniz's series: Pi/4 = Sum_{n=0..inf} (-1)^n/(2n+1) (cf. A072172).
%C A005408 Beginning of the ordering of the natural numbers used in Sharkovski's
theorem - see the Cielsielski-Pogoda paper.
%C A005408 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 A005408 Apart from initial term(s), dimension of the space of weight 2n cusp
forms for Gamma_0( 6 ).
%C A005408 Also continued fraction for coth(1) (A073747 is decimal expansion). -
Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 07 2002
%C A005408 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
%C A005408 Smallest number greater than n, not a multiple of n, but containing it
in binary representation. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 06 2003
%C A005408 Numbers n such that phi(2n)=phi(n), where phi is the Euler's totient(A000010).
- Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 27 2004
%C A005408 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 (Gerald.McGarvey(AT)comcast.net), Feb 04 2005
%C A005408 a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Jun 01 2005
%C A005408 For n>1, numbers having 2 as an anti-divisor. - Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be),
Oct 02 2005
%C A005408 a(n) = shortest side a of all integer-sided triangles with sides a<=b<=c
and inradius n >= 1.
%C A005408 First differences of squares (A000290). - Lekraj Beedassy (blekraj(AT)yahoo.com),
Jul 15 2006
%C A005408 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 A005408 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 A005408 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
(reinhard.zumkeller(AT)gmail.com), Mar 04 2007
%C A005408 For n > 2, a(n-1) is the least integer not the sum of < n n-gonal numbers
(0 allowed). - Jonathan Sondow (jsondow(AT)alumni.princeton.edu),
Jul 01 2007
%C A005408 A134451(a(n)) = ABS(A134452(a(n))) = 1; union of A134453 and A134454.
- Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 27 2007
%C A005408 Numbers n such that sigma(2n)=3*sigma(n). - Farideh Firoozbakht (mymontain(AT)yahoo.com),
Feb 26 2008
%C A005408 a(n) = A139391(A016825(n)) = A006370(A016825(n)). - Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Apr 17 2008
%C A005408 a(n)=a(n-1)+A091137(1). First sequence of a family. Generally first term
is A091137(n+1)/2. Second sequence: A017593; third: A073762; ...
Differences give A007395. - Paul Curtz (bpcrtz(AT)free.fr), Jul 15
2008
%C A005408 Number of divisors of 4^n - J. Lowell (jhbubby(AT)mindspring.com), Aug
30 2008
%C A005408 Equals INVERT transform of A078050 (signed - Cf. comments); and row sums
of triangle A144106. [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 11 2008]
%C A005408 odd numbers(n) = 2*n+1 = square pyramidal number(3*n+1) / triangular
number(3*n+1) [From Pierre CAMI (pierrecami(AT)tele2.fr), Sep 27
2008]
%C A005408 A000035(a(n))=1, A059841(a(n))=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Sep 29 2008]
%C A005408 Multiplicative closure of A065091. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 14 2008]
%C A005408 Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Oct 28 2008:
(Start)
%C A005408 (APSO) Alternating partial sums of
%C A005408 (a-b+c-d+e-f+g...)=(a+b+c+d+e+f+g...)-2*(b+d+f...)
%C A005408 it appears that APSO A005408 =
%C A005408 A131738 = A000290 - 2*(A014105 repeated)
%C A005408 (End)
%C A005408 Let n>1; if a=n, b=(n^2-1)/2, c=(n^2+1)/2, then a^2+b^2=c^2 is alwais
verified. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Mar 27 2009]
%C A005408 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 (surianonoi5(AT)libero.it), Jun 08 2009]
%C A005408 Except for the first term, numbers n such that if A=(7*n^4-3*n^3-3*n^2-n)/
8; B=(7*n^4+4*n^3-6*n^2-4*n-1)/16; C=(17*n^4+20*n^3+6*n^2+4*n+1)/
16; D=(5*n^4+3*n^3+3*n^2+n)/4; then A^3+B^3+C^3=D^3 [From Vincenzo
Librandi (vincenzo.librandi(AT)tin.it), Sep 08 2009]
%C A005408 Binomial transform of A130706, inverse binomial transform of A001787(without
the initial 0). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 17 2009]
%C A005408 Also the 3-rough numbers: positive integers that have no prime factors
less than 3. [From Michael Porter (michael_b_porter(AT)yahoo.com),
Oct 08 2009]
%C A005408 0r n without 2 as prime factor. [From Juri-StepanGerasimov (2stepan(AT)rambler.ru),
Nov 19 2009]
%D A005408 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A005408 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 2.
%D A005408 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A005408 Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer
Sequences, Vol. 9 (2006), Article 06.2.3.
%D A005408 K. Ciesielski and Z. Pogoda, On ordering the natural numbers, or the
Sharkovski theorem, Amer. Math. Monthly, 115 (No. 2, 2008), 158-165.
%D A005408 T. Dantzig, The Language of Science, 4th Edition (1954) page 276.
%D A005408 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY,
1965, p. 73.
%D A005408 D. Hoek, Parvisa moenster i permutationer [Swedish], (2007).
%D A005408 E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998,
pp. 203-205.
%H A005408 N. J. A. Sloane, <a href="b005408.txt">Table of n, a(n) for n = 0..10000</
a>
%H A005408 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A005408 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=935">
Encyclopedia of Combinatorial Structures 935</a>
%H A005408 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A005408 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A005408 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005408 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0n.gp">
Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A005408 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The
modular forms database</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
OddNumber.html">Link to a section of The World of Mathematics.</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Davenport-SchinzelSequence.html">Link to a section of The World of
Mathematics.</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GnomonicNumber.html">Link to a section of The World of Mathematics.</
a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PythagoreanTriple.html">Link to a section of The World of Mathematics.</
a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
InverseTangent.html">Inverse Tangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
InverseCotangent.html">Inverse Cotangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
InverseHyperbolicCotangent.html">Inverse Hyperbolic Cotangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
InverseHyperbolicTangent.html">Inverse Hyperbolic Tangent</a>
%H A005408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
NexusNumber.html">Nexus Number</a>
%H A005408 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A005408 Euler transform of length 2 sequence [ 3, -1]. - Michael Somos Mar 30
2007
%F A005408 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 A005408 a(n)= 2*n+1 . a(-1-n)= -a(n) . a(n+1)= a(n)+2 .
%F A005408 G.f.: (1+x)/ (1-x)^2 . E.g.f.: (1+2*x)* exp(x) .
%F A005408 a(n)= b(2*n+1) where b(n)= n is multiplicative.
%F A005408 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
%F A005408 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
%F A005408 A005408(n)=A000330(A016777(n))/A000217(A016777(n)) [From Pierre CAMI
(pierrecami(AT)tele2.fr), Sep 27 2008]
%F A005408 a(n)=2*a(n-1)-a(n-2); a(0)=1, a(1)=3. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 03 2008]
%F A005408 a(n) = A034856(n+1) - A000217(n) = A005843(n) + A000124(n) - A000217(n)
= A005843(n) + 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz),
Sep 05 2009]
%F A005408 a(n)=4*n-a(n-1)-4 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Nov 22 2009]
%e A005408 For n=3, A=57, B=38, C=124, D=129, and 57^3+38^3+124^3=129^3; For n=5,
A=490, B=294, C=831, D=895, and 490^3+294^3+831^3=895^3 [From Vincenzo
Librandi (vincenzo.librandi(AT)tin.it), Sep 08 2009]
%e A005408 For n=2, a(2)=4*2-1-4=3; n=3, a(3)=4*3-3-4=5; n=4, a(4)=4*4-5-4=7 [From
Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009]
%p A005408 A005408 := n->2*n+1;
%p A005408 A005408:=(1+z)/(z-1)^2; [S. Plouffe in his 1992 dissertation.]
%p A005408 a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+2 od: seq(a[n], n=1..66);
- Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
%t A005408 Table[2n - 1, {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com),
Apr 01 2006
%o A005408 (MAGMA) [ 2*n+1 : n in [0..100]];
%o A005408 (PARI) {a(n)= 2*n+1}
%o A005408 (Other) [i for i in range(142) if gcd(4,i) == 1] [From Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Apr 21 2009]
%o A005408 (Other) sage: [2* bernoulli_polynomial(n,1) for n in xrange(1, 67)]#
[From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
%Y A005408 Cf. A000027, A005843.
%Y A005408 See A120062 for sequences related to integer-sided triangles with integer
inradius n.
%Y A005408 Cf. A128200, A000290.
%Y A005408 A078050, A144106 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11
2008]
%Y A005408 Sequence in context: A053229 A157142 A004273 this_sequence A144396 A060747
A089684
%Y A005408 Adjacent sequences: A005405 A005406 A005407 this_sequence A005409 A005410
A005411
%K A005408 easy,nonn,core,nice,new
%O A005408 0,2
%A A005408 N. J. A. Sloane (njas(AT)research.att.com).
%E A005408 More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 06 2003
%E A005408 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Mar 11 2009
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