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A004273 0 together with odd numbers. 28
0, 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

0,3

COMMENTS

Also continued fraction for tanh(1) (A073744 is decimal expansion). - Rick L. Shepherd, Aug 07 2002

From Alexander R. Povolotsky and _Paolo Lava_, Oct 29 2009: (Start)

From Inverse Symbolic Calculator Plus http://glooscap.cs.dal.ca:8087/advancedCalc Advanced lookup results for sum(2/(2^(n+1))/GAMMA(n+1/2)*Pi^(1/2),n = 1 .. infinity) Transform Searched for Description K*1 1.4106861346424479976908247 Sum(1/prod(A004273(k),k=1..n),n=1..inf) Below are two Maple programs, developed by Paolo Lava confirming that indeed sum(2/(2^(n+1))/GAMMA(n+1/2)*Pi^(1/2),n = 1 .. infinity) = Sum(1/prod(A004273(k),k=1..n),n=1..inf). To reiterate, it appears that indeed the two formulas practically give the same result!

Maple program for Sum(1/prod(A004273(k),k=1..n),n=1..inf)is:

Formula1:=proc(i) local a,k,n,t; for n from 1 by 1 to i do a:=add(1/product(2*t-1,t=1..k),k=1..n); print(evalf(a,600)); od; end: Formula1(10000);

Maple program for the formula using GAMMA function is:

Formula2:=proc(i) local a,k,n; for n from 1 by 1 to i do a:=add(2/(2^(k+1))/GAMMA(k+1/2)*Pi^(1/2),k=1..n); print(evalf(a,600)); od; end: Formula2(10000); Both programs were run up to 10.000 iterations showing 599 decimal digits.

The result in both cases is: 1.41068613464244799769082471141911504132347\

862562519219772463946816478179849039792711540922477861164014728970035593\

291934262239437689612130677631195100435759045028697694516138268925799622\

506579245758816483482960481133594351367886637443783678748021144275108269\

196477247936726250874958337834244668843998292968423370781551842367181745\

798283956182034092760339072832832252093637885530596099628134118249573271\

812709090115944540248304702415273410481321124791326873921867111910022107\

760939194553035779605182699929996414630218895949183315671171755021724947\

333256207314724810499711097293803256333031250513313069 (End)

From Jaroslav Krizek, May 28 2010: (Start)

For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is integer. A040001(a(n)) = 1. See A145051 and A040001.

For n >= 1, a(n) = corresponding values of antiharmonic means to numbers from A016777 (numbers k such that antiharmonic mean of the first k positive integers is integer).

a(n) = A000330(A016777(n)) / A000217(A016777(n)) = A146535(A016777(n)+1). (End)

If the n-th prime is denoted by p(n) then it appears that a(j) = distinct, increasing values of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) for each j. - Christopher Hunt Gribble, Oct 05 2010

A214546(a(n)) > 0. - Reinhard Zumkeller, Jul 20 2012

Dimension of the space of weight 2n+2 cusp forms for Gamma_0(6).

LINKS

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

FORMULA

a(n) = 2*n-[(n+2) mod (n+1)], with n>=0. - Paolo P. Lava, Aug 29 2007

G.f.: x*(1+x)/(-1+x)^2. - R. J. Mathar, Nov 18 2007

a(n) = lodumo_2(A057427(n)). - Philippe Deléham, Apr 26 2009

Euler transform of length 2 sequence [ 3, -1]. - Michael Somos, Jul 03 2014

a(n) = (4*n-1-(-1)^(2^n))/2. - Luce ETIENNE, Jul 11 2015

EXAMPLE

G.f. = x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 + 11*x^6 + 13*x^7 + 15*x^8 + 17*x^9 + ...

MAPLE

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+2 od: seq(a[n], n=0..66); # Zerinvary Lajos, Mar 16 2008

MATHEMATICA

Join[{0}, Range[1, 200, 2]] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

PROG

(MAGMA) [2*n-Floor((n+2) mod (n+1)): n in [0..70]]; // Vincenzo Librandi, Sep 21 2011

(Sage) def a(n) : return( dimension_cusp_forms( Gamma0(6), 2*n+2) ); # Michael Somos, Jul 03 2014

(PARI) a(n)=max(2*n-1, 1) \\ Charles R Greathouse IV, May 14 2014

CROSSREFS

Cf. A110185, continued fraction expansion of 2*tanh(1/2), and A204877, continued fraction expansion of 3*tanh(1/3). [Bruno Berselli, Jan 26 2012]

Cf. A005408.

Sequence in context: A247328 * A005408 A176271 A144396 A060747 A089684

Adjacent sequences:  A004270 A004271 A004272 * A004274 A004275 A004276

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 21 19:52 EDT 2017. Contains 290906 sequences.