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A133694
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a(n) = (3*n^2 + 3*n - 4)/2.
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7
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1, 7, 16, 28, 43, 61, 82, 106, 133, 163, 196, 232, 271, 313, 358, 406, 457, 511, 568, 628, 691, 757, 826, 898, 973, 1051, 1132, 1216, 1303, 1393, 1486, 1582, 1681, 1783, 1888, 1996, 2107, 2221, 2338, 2458, 2581, 2707, 2836, 2968, 3103, 3241, 3382, 3526, 3673
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OFFSET
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1,2
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COMMENTS
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Binomial transform of 1, 6, 3 followed by A000004, i.e., 1, 6, 3, 0, 0, 0, 0, ... .
Equals (1, 2, 3, 4, ...) convolved with (1, 5, 3, 3, 3, ...). Example: a(4) = (1, 2, 3, 4) dot (3, 3, 5, 1) = (3 + 6 + 15 + 4) = 28. - Gary W. Adamson, May 01 2009
Equivalently, numbers of the form 3*(h+1)*(2*h-1) + 1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 03 2017
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = 1/2 + 2*Pi*tan(sqrt(19/3)*Pi/2)/sqrt(57). - Amiram Eldar, Jun 08 2022
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EXAMPLE
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a(3) = 3*A000217(3) - 2 = 3*6 - 2 = 16.
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MAPLE
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MATHEMATICA
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Table[(3*n^2 + 3*n - 4)/2, {n, 100}]
CoefficientList[Series[(1 + 4 x - 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 30 2014 *)
LinearRecurrence[{3, -3, 1}, {1, 7, 16}, 50] (* Harvey P. Dale, Sep 05 2020 *)
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PROG
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(Magma) a000217:=func<n | n*(n+1) div 2>; [3*a000217(n)-2: n in [1..60]];
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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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