This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211775 #55 Feb 16 2025 08:33:17
%S A211775 5419,5209,5003,4801,4603,4409,4219,4033,3851,3673,3499,3329,3163,
%T A211775 3001,2843,2689,2539,2393,2251,2113,1979,1849,1723,1601,1483,1369,
%U A211775 1259,1153,1051,953,859,769,683,601,523,449,379,313,251,193,139,89,43,1,-37,-71,-101,-127
%N A211775 a(n) = 2*n^2 - 212*n + 5419.
%C A211775 A "prime-generating" polynomial: This polynomial generates 92 primes (57 being distinct) for 0 <= n <= 99 (in fact the next seven terms are still primes but we keep the range 0-99, customary for comparisons), just three primes fewer than the record held by Euler's polynomial for n = m - 35, which is m^2 - 69*m + 1231 (see the link below).
%C A211775 The nonprime terms in the first 100 are 1, 1369 = 37^2, 1849 = 43^2, 4033 = 37*109 (all taken twice).
%C A211775 Setting n = 2*m + 54 we obtain the polynomial 8*m^2 + 8*m - 197, which generates 31 primes in a row starting from m = 0 (the polynomial 8*m^2 - 488*m + 7243 generates the same 31 primes, but in reverse order).
%C A211775 The substitution n = m + 53 converts this polynomial to the simpler form 2*m^2 - 199. - _Charles Kusniec_, Nov 11 2016
%D A211775 Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.
%H A211775 Bruno Berselli, <a href="/A211775/b211775.txt">Table of n, a(n) for n = 0..1000</a>
%H A211775 Joe L. Mott and Kermite Rose, <a href="https://www.math.fsu.edu/~aluffi/archive/paper134.ps">Prime-Producing Cubic Polynomials</a>.
%H A211775 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.
%H A211775 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A211775 G.f.: (5419 - 11048*x + 5633*x^2)/(1-x)^3. - _Bruno Berselli_, May 18 2012
%F A211775 From _Elmo R. Oliveira_, Feb 09 2025: (Start)
%F A211775 E.g.f.: exp(x)*(5419 - 210*x + 2*x^2).
%F A211775 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%p A211775 A211775:=n->2*n^2 - 212*n + 5419: seq(A211775(n), n=0..100); # _Wesley Ivan Hurt_, Jan 20 2017
%t A211775 Table[2*n^2 - 212*n + 5419, {n, 0, 80}] (* _Wesley Ivan Hurt_, Aug 06 2017 *)
%o A211775 (Magma) [2*n^2-212*n+5419: n in [0..49]]; // _Bruno Berselli_, May 18 2012
%o A211775 (PARI) Vec((5419-11048*x+5633*x^2)/(1-x)^3+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%o A211775 (PARI) a(n) = 2*n^2 - 212*n + 5419 \\ _Charles R Greathouse IV_, Dec 19 2016
%K A211775 sign,easy
%O A211775 0,1
%A A211775 _Marius Coman_, May 18 2012
%E A211775 Edited by _N. J. A. Sloane_, Nov 12 2016