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A004273
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0 together with odd numbers.
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31
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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
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OFFSET
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0,3
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COMMENTS
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Also continued fraction for tanh(1) (A073744 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
From Alexander R. Povolotsky and Paolo P. 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 10000 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).
The size of a maximal 2-degenerate graph of order n-1 (this class includes 2-trees and maximal outerplanar graphs (MOPs)). - Allan Bickle, Nov 14 2021
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..10000
A. Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.
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FORMULA
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a(n) = 2*n - ((n+2) mod (n+1)), 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
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EXAMPLE
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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 + ...
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MAPLE
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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
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MATHEMATICA
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Join[{0}, Range[1, 200, 2]] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
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PROG
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(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, n) \\ Charles R Greathouse IV, May 14 2014
(GAP) Concatenation([0], List([1, 3..141])); # Muniru A Asiru, Jul 28 2018
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CROSSREFS
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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 A317107 A317439 * A005408 A176271 A144396
Adjacent sequences: A004270 A004271 A004272 * A004274 A004275 A004276
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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