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%I #56 Feb 12 2024 02:28:21
%S 0,8,18,30,44,60,78,98,120,144,170,198,228,260,294,330,368,408,450,
%T 494,540,588,638,690,744,800,858,918,980,1044,1110,1178,1248,1320,
%U 1394,1470,1548,1628,1710,1794,1880,1968,2058,2150,2244,2340,2438,2538,2640,2744,2850,2958,3068,3180,3294
%N a(n) = n*(n+7).
%C 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)
%H Vincenzo Librandi, <a href="/A028563/b028563.txt">Table of n, a(n) for n = 0..1000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>.
%H Felix P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, Preprint on ResearchGate, March 2014.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F Equals 2*A055999. - _Zerinvary Lajos_, Feb 12 2007
%F a(n) = 2*n+a(n-1)+6. - _Vincenzo Librandi_, Aug 05 2010
%F Sum_{n>=1} 1/a(n) = 363/980 = 0.37040816... - _R. J. Mathar_, Mar 22 2011
%F G.f.: 2*x*(4-3*x)/(1-x)^3. - _Colin Barker_, Feb 17 2012
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/7 - 319/2940. - _Amiram Eldar_, Jan 15 2021
%F From _Amiram Eldar_, Feb 12 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = 720*cos(sqrt(53)*Pi/2)/(143*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = -112*cos(3*sqrt(5)*Pi/2)/(11*Pi). (End)
%p A028563:=n->n*(n + 7); seq(A028563(n), n=0..50); # _Wesley Ivan Hurt_, Mar 30 2014
%t CoefficientList[Series[2 x (4 - 3 x)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 18 2013 *)
%t LinearRecurrence[{3,-3,1},{0,8,18},60] (* _Harvey P. Dale_, Oct 07 2015 *)
%o (Magma) [n*(n+7): n in [0..60]]; // _Vincenzo Librandi_, Oct 18 2013
%o (PARI) a(n)=n*(n+7) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A002522, A055999.
%K nonn,easy
%O 0,2
%A _Patrick De Geest_
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