A028881 a(n) = n^2 - 7.

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

Author

Patrick De Geest

Keywords

• nonn
• easy

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).

Programs

Mathematica

LinearRecurrence[{3,-3,1}, {2,9,18}, 50] (* G. C. Greubel, Aug 19 2017 *)

PARI

a(n)=n^2-7 \\ Charles R Greathouse IV, Oct 07 2015

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)