OFFSET
3,1
COMMENTS
a(n), n>=0, with a(0) = -7, a(1) = -6 and a(2) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 28 for b = 2*n. 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 16 2013
The product of two consecutive terms belongs to the sequence. - Klaus Purath, Dec 13 2022 [a(n)*a(n+1) = a(n^2 + n - 7). - Wolfdieter Lang, Dec 15 2022]
LINKS
G. C. Greubel, Table of n, a(n) for n = 3..1000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = a(n-1) + 2*n - 1, with a(3)=2. - Vincenzo Librandi, Aug 05 2010
G.f.: x^3*(2+3*x-3*x^2)/(1-x)^3. - Colin Barker, Feb 17 2012
E.g.f.: (1/2)*(2*(x^2 + x -7)*exp(x) + 14 + 12*x + 3*x^2). - G. C. Greubel, Aug 19 2017
From Amiram Eldar, Nov 04 2020: (Start)
Sum_{n>=3} 1/a(n) = (8 - sqrt(7)*Pi*cot(sqrt(7)*Pi))/14.
Sum_{n>=3} (-1)^(n+1)/a(n) = (-10 + 3*sqrt(7)*Pi*cosec(sqrt(7)*Pi))/42. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=3} (1 - 1/a(n)) = (9/(4*sqrt(14)))*sin(2*sqrt(2)*Pi)/sin(sqrt(7)*Pi).
Product_{n>=3} (1 + 1/a(n)) = (3*sqrt(21/2)/5)*sin(sqrt(6)*Pi)/sin(sqrt(7)*Pi). (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {2, 9, 18}, 50] (* G. C. Greubel, Aug 19 2017 *)
PROG
(PARI) a(n)=n^2-7 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved