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A028563
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a(n) = n*(n+7).
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21
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0, 8, 18, 30, 44, 60, 78, 98, 120, 144, 170, 198, 228, 260, 294, 330, 368, 408, 450, 494, 540, 588, 638, 690, 744, 800, 858, 918, 980, 1044, 1110, 1178, 1248, 1320, 1394, 1470, 1548, 1628, 1710, 1794, 1880, 1968, 2058, 2150, 2244, 2340, 2438, 2538, 2640, 2744, 2850, 2958, 3068, 3180, 3294
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OFFSET
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0,2
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COMMENTS
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a(m), for m >= 1, are the only positive integer values of t for which the Binet-de Moivre formula of the recurrence b(n) = 7*b(n-1)+t*b(n-2) has a square root whose radicand is a square. In particular, sqrt(7^2+4*t) is a positive integer since 7^2+4*t = 7^2+4*a(m) = (2*m+7)^2. Thus the characteristic roots are r1 = 7+m and r2 = -m. - Felix P. Muga II, Mar 28 2014 (edited - Wolfdieter Lang, Apr 17 2014)
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = 363/980 = 0.37040816... - R. J. Mathar, Mar 22 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/7 - 319/2940. - Amiram Eldar, Jan 15 2021
Product_{n>=1} (1 - 1/a(n)) = 720*cos(sqrt(53)*Pi/2)/(143*Pi).
Product_{n>=1} (1 + 1/a(n)) = -112*cos(3*sqrt(5)*Pi/2)/(11*Pi). (End)
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MAPLE
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MATHEMATICA
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CoefficientList[Series[2 x (4 - 3 x)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Oct 18 2013 *)
LinearRecurrence[{3, -3, 1}, {0, 8, 18}, 60] (* Harvey P. Dale, Oct 07 2015 *)
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PROG
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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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STATUS
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approved
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