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A256833
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a(n) = (4*n+3)*(4*n+2).
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1
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6, 42, 110, 210, 342, 506, 702, 930, 1190, 1482, 1806, 2162, 2550, 2970, 3422, 3906, 4422, 4970, 5550, 6162, 6806, 7482, 8190, 8930, 9702, 10506, 11342, 12210, 13110, 14042, 15006, 16002, 17030, 18090, 19182, 20306, 21462, 22650, 23870, 25122, 26406
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OFFSET
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0,1
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COMMENTS
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Since 0 = Sin(Pi) = Sum_{n>=0}(-1)^n*Pi^(2n+1)/(2n+1)!, we can move the negative terms to the other side of the equation to get: Sum_{n>=0} Pi^(4n+1)/(4n+1)! = Sum_{n>=0}Pi^(4n+3)/(4n+3)!.
Now, if we let f(n) = Pi^(4n+1)/(4n+1)!, then the previous equation can be written as Sum_{n>=0}f(n) = Sum_{n>=0}(Pi^2/((4*n+3)*(4*n+2)))*f(n); a(n) is the n-th denominator on the right hand side.
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LINKS
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FORMULA
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a(n) = 16*n^2 + 20*n + 6.
Sum_{n>=0} 1/a(n) = Pi/8 - log(2)/4.
Sum_{n>=0} (-1)^n/a(n) = sqrt(2)*log(sqrt(2)+1)/4 - (sqrt(2)-1)*Pi/8. (End)
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MATHEMATICA
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CoefficientList[Series[(6 + 24 x + 2 x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 12 2015 *)
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PROG
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(PARI) vector(50, n, (4*n-1)*(4*n-2)) \\ Derek Orr, Apr 13 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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