0,2

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 (rshepherd2(AT)hotmail.com), 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 (reinhard.zumkeller(AT)gmail.com), Oct 06 2003

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

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

a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (blekraj(AT)yahoo.com), 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 (reinhard.zumkeller(AT)gmail.com), 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 (jsondow(AT)alumni.princeton.edu), Jul 01 2007

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

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

a(n) = A139391(A016825(n)) = A006370(A016825(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 17 2008

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

Number of divisors of 4^n - 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 (qntmpkt(AT)yahoo.com), Sep 11 2008]

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]

A000035(a(n))=1, A059841(a(n))=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 29 2008]

Multiplicative closure of A065091. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 14 2008]

Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Oct 28 2008: (Start)

(APSO) Alternating partial sums of

(a-b+c-d+e-f+g...)=(a+b+c+d+e+f+g...)-2*(b+d+f...)

it appears that APSO A005408 =

A131738 = A000290 - 2*(A014105 repeated)

(End)

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]

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]

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]

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 (michael_b_porter(AT)yahoo.com), Oct 08 2009]

0r n without 2 as prime factor. [From Juri-StepanGerasimov (2stepan(AT)rambler.ru), Nov 19 2009]

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

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, Table of n, a(n) for n = 0..10000

Index entries for sequences related to linear recurrences with constant coefficients

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.

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Inverse Tangent

Eric Weisstein's World of Mathematics, Inverse Cotangent

Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cotangent

Eric Weisstein's World of Mathematics, Inverse Hyperbolic Tangent

Eric Weisstein's World of Mathematics, Nexus Number

Index entries for "core" sequences

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)= 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) .

a(n)= b(2*n+1) where b(n)= n is multiplicative.

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 (pierrecami(AT)tele2.fr), Sep 27 2008]

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]

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]

a(n)=4*n-a(n-1)-4 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009]

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]

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]

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

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

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

Table[2n - 1, {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006

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

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

(Other) [i for i in range(142) if gcd(4, i) == 1] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2009]

(Other) sage: [2* bernoulli_polynomial(n, 1) for n in xrange(1, 67)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]

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 (qntmpkt(AT)yahoo.com), Sep 11 2008]

Sequence in context: A053229 A157142 A004273 this_sequence A144396 A060747 A089684

Adjacent sequences: A005405 A005406 A005407 this_sequence A005409 A005410 A005411

easy,nonn,core,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 06 2003

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009