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a(n) = 2*n^2 - 4*n + 73.
1

%I #38 Nov 09 2024 02:23:04

%S 71,73,79,89,103,121,143,169,199,233,271,313,359,409,463,521,583,649,

%T 719,793,871,953,1039,1129,1223,1321,1423,1529,1639,1753,1871,1993,

%U 2119,2249,2383,2521,2663,2809,2959,3113,3271,3433,3599,3769,3943,4121,4303,4489

%N a(n) = 2*n^2 - 4*n + 73.

%C Extrapolates a quadratic passing through 71, 73, and 79.

%H Michael M. Ross, <a href="http://naturalnumbers.org/polyprimes.html">Poly Primes: The Polynomiality of Proximate Primes</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: x*(71 - 140*x + 73*x^2)/(1 - x)^3. - _Arkadiusz Wesolowski_, Oct 24 2013

%F Sum_{n>=1} 1/a(n) = 1/142 + coth(sqrt(71/2)*Pi)/(2*sqrt(142)). - _Amiram Eldar_, Jul 30 2024

%F From _Elmo R. Oliveira_, Nov 03 2024: (Start)

%F E.g.f.: exp(x)*(2*x^2 - 2*x + 73) - 73.

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

%e If n=10 then 2*n^2 - 4*n + 73 = 233.

%t Table[2*n^2 - 4*n + 73, {n, 48}] (* _Arkadiusz Wesolowski_, Oct 24 2013 *)

%o (Magma) [2*n^2-4*n+73 : n in [1..48]]; // _Arkadiusz Wesolowski_, Oct 24 2013

%o (PARI) vector(48, n, 2*n^2-4*n+73) \\ _Arkadiusz Wesolowski_, Oct 24 2013

%Y Cf. A126665, A126719.

%K easy,nonn

%O 1,1

%A _Michael M. Ross_, Mar 28 2007

%E Extended by _Charles R Greathouse IV_, Jul 25 2010