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A028881
a(n) = n^2 - 7.
8
2, 9, 18, 29, 42, 57, 74, 93, 114, 137, 162, 189, 218, 249, 282, 317, 354, 393, 434, 477, 522, 569, 618, 669, 722, 777, 834, 893, 954, 1017, 1082, 1149, 1218, 1289, 1362, 1437, 1514, 1593, 1674, 1757, 1842, 1929, 2018, 2109, 2202, 2297, 2394
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]
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
Sequence in context: A282519 A359580 A103256 * A294535 A294543 A295956
KEYWORD
nonn,easy
STATUS
approved