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A014209
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a(n) = n^2 + 3*n - 1.
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26
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-1, 3, 9, 17, 27, 39, 53, 69, 87, 107, 129, 153, 179, 207, 237, 269, 303, 339, 377, 417, 459, 503, 549, 597, 647, 699, 753, 809, 867, 927, 989, 1053, 1119, 1187, 1257, 1329, 1403, 1479, 1557, 1637, 1719, 1803, 1889, 1977, 2067, 2159, 2253, 2349, 2447, 2547, 2649
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OFFSET
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0,2
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COMMENTS
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Difference between n-th centered hexagonal number and (2n)^2. - Alonso del Arte, Jul 06 2004
Given the roots to n^2 + 3n - 1, a = -3.302775..., b = 0.302775...; then a(n) = (n + 3 + a) * (n + 3 + b). Example: a(3) = 17 = (6 - 3.302...) * (6 + 0.302775) - Gary W. Adamson, Jul 29 2009
a(n-1) = n*(n+1) - 3, with a(-1) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 13 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
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LINKS
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FORMULA
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a(n) = 3*a(n-1)-3*a(n-2)+a(n-3).
G.f.: (-1+6*x-3*x^2)/(1-x)^3. (End)
Sum_{n>=0} 1/a(n) = 1/3 + tan(sqrt(13)*Pi/2)*Pi/sqrt(13). - Amiram Eldar, Jan 08 2023
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MATHEMATICA
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CoefficientList[Series[(- 1 + 6 x - 3 x^2)/(1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 15 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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