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A004273
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0 together with odd numbers.
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18
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also continued fraction for tanh(1) (A073744 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
Lodumo_2 of A057427 . [From Philippe DELEHAM, Apr 26 2009]
Contribution 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 formulae 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 of the contribution from Alexander R. Povolotsky and Paolo Lava)
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. [From Jaroslav Krizek, May 28 2010]
Contribution from Jaroslav Krizek, May 28 2010: (Start)
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
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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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)=lod_2(A057427(n)). [From Philippe DELEHAM, Apr 26 2009]
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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 (zerinvarylajos(AT)yahoo.com), Mar 16 2008
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PROG
| (MAGMA) [2*n-Floor((n+2) mod (n+1)): n in [0..70]]; // Vincenzo Librandi, Sep 21 2011
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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
Sequence in context: A157142 * A005408 A176271 A144396 A060747 A089684
Adjacent sequences: A004270 A004271 A004272 * A004274 A004275 A004276
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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