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 A210626 #55 Feb 16 2025 08:33:17
%S A210626 3449,3169,2897,2633,2377,2129,1889,1657,1433,1217,1009,809,617,433,
%T A210626 257,89,-71,-223,-367,-503,-631,-751,-863,-967,-1063,-1151,-1231,
%U A210626 -1303,-1367,-1423,-1471,-1511,-1543,-1567,-1583,-1591,-1591,-1583,-1567,-1543,-1511,-1471
%N A210626 Values of the prime-generating polynomial 4*n^2 - 284*n + 3449.
%C A210626 The polynomial successively generates 35 primes/negative values of primes starting at n = 0.
%C A210626 This polynomial generates 95 primes in absolute value (60 distinct ones) for n from 0 to 99, equaling the record held by Euler's polynomial for n = m - 35, which is m^2 - 69*m + 1231 (see the reference).
%C A210626 The nonprime terms (in absolute value) up to n = 99 are: 1591 = 37*43, 3737 = 37*101, 4033 = 37*109; 5633 = 43*131; 5977 = 43*139; 9017 = 71*127.
%C A210626 The polynomial 4*n^2 + 12*n - 1583 generates the same 35 primes in row starting from n = 0 in reverse order.
%C A210626 Note: in the same family of prime-generating polynomials (with the discriminant equal to 199*2^p, where p is odd) there are the polynomial 32*n^2 - 944*n + 6763 (with its "reversed polynomial" 32*m^2 - 976*m + 7243, for m=30-n), generating 31 primes in row, and the polynomial 4*n^2 - 428*n + 5081 (with 4*m^2 + 188*m - 4159, for m = 30 - n), generating 31 primes in row.
%D A210626 Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials, Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.
%H A210626 Vincenzo Librandi, <a href="/A210626/b210626.txt">Table of n, a(n) for n = 0..1000</a>
%H A210626 Eric Weisstein's World of Mathematic, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.
%H A210626 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A210626 G.f.: (3449 - 7178*x + 3737*x^2)/(1-x)^3. - _Bruno Berselli_, Jun 07 2012
%F A210626 From _Elmo R. Oliveira_, Feb 09 2025: (Start)
%F A210626 E.g.f.: exp(x)*(3449 - 280*x + 4*x^2).
%F A210626 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A210626 LinearRecurrence[{3,-3, 1},{3449,3169,2897},100] (* _Vincenzo Librandi_, Aug 01 2012 *)
%o A210626 (PARI) Vec((3449-7178*x+3737*x^2)/(1-x)^3+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%K A210626 sign,easy
%O A210626 0,1
%A A210626 _Marius Coman_, May 08 2012