A126665 a(n) = -n^2 + 9*n + 53.

53, 61, 67, 71, 73, 73, 71, 67, 61, 53, 43, 31, 17, 1, -17, -37, -59, -83, -109, -137, -167, -199, -233, -269, -307, -347, -389, -433, -479, -527, -577, -629, -683, -739, -797, -857, -919, -983, -1049, -1117, -1187, -1259, -1333, -1409, -1487, -1567, -1649, -1733, -1819, -1907, -1997, -2089, -2183, -2279

Offset: 0

Author

Michael M. Ross, Mar 13 2007

Keywords

• sign
• easy

Comments

Quadratic equation derived from the four primes 61, 67, 71, 73 using the method of common differences. Many of the initial terms are primes.

Examples

			For n=8, -1*8^2 + 9*8 + 53 = 61.
		

Links

• Michael M. Ross, Natural Numbers.
• Robert Sacks, Number Spiral: Method of Common Differences.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A186950.

Programs

Magma

[-n^2+9*n+53 : n in [0..46]]; // Arkadiusz Wesolowski, Oct 24 2013

Mathematica

Table[ - n^2 + 9*n + 53, {n, 0, 46}] (* Arkadiusz Wesolowski, Oct 24 2013 *)
LinearRecurrence[{3,-3,1},{53,61,67},60] (* Harvey P. Dale, Apr 04 2024 *)

PARI

a(n) = -n^2 + 9*n + 53 \\ Michel Marcus, Jun 30 2013

Formula

From Arkadiusz Wesolowski, Oct 24 2013: (Start)

a(n) = -A186950(n+19).

G.f.: (53 - 98*x + 43*x^2)/(1 - x)^3. (End)

From Elmo R. Oliveira, Nov 02 2024: (Start)

E.g.f.: (53 + 8*x - x^2)*exp(x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)