OFFSET
0,2
COMMENTS
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)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 2*A055999(n). - Zerinvary Lajos, Feb 12 2007
a(n) = 2*n + a(n-1) + 6. - Vincenzo Librandi, Aug 05 2010
Sum_{n>=1} 1/a(n) = 363/980 = 0.37040816... - R. J. Mathar, Mar 22 2011
G.f.: 2*x*(4-3*x)/(1-x)^3. - Colin Barker, Feb 17 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/7 - 319/2940. - Amiram Eldar, Jan 15 2021
From Amiram Eldar, Feb 12 2024: (Start)
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)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: exp(x)*x*(8 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
MAPLE
MATHEMATICA
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 *)
PROG
(Magma) [n*(n+7): n in [0..60]]; // Vincenzo Librandi, Oct 18 2013
(PARI) a(n)=n*(n+7) \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved