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 A181963 #49 Feb 09 2025 13:18:06
%S A181963 14083,12923,11813,10753,9743,8783,7873,7013,6203,5443,4733,4073,3463,
%T A181963 2903,2393,1933,1523,1163,853,593,383,223,113,53,43,83,173,313,503,
%U A181963 743,1033,1373,1763,2203,2693,3233,3823,4463,5153,5893,6683,7523,8413,9353,10343,11383,12473
%N A181963 Prime-generating polynomial: a(n) = 25*n^2 - 1185*n + 14083.
%C A181963 The polynomial generates 32 primes starting from n = 0.
%C A181963 The polynomial 25*n^2 - 365*n + 1373 generates the same primes in reverse order.
%C A181963 This family of prime-generating polynomials (with the discriminant equal to -4075 = -163*5^2) is interesting for generating primes of same form: the polynomial 25*n^2 - 395*n + 1601 generates 16 primes of the form 10*k + 1 (1601, 1231, 911, 641, 421, 251, 131, 61, 41, 71, 151, 281, 461, 691, 971, 1301) and the polynomial 25*n^2 + 25*n + 47 generates 16 primes of the form 10*k + 7 (47, 97, 197, 347, 547, 797, 1097, 1447, 1847, 2297, 2797, 3347, 3947, 4597, 5297, 6047).
%C A181963 Note: all the polynomials of the form 25*n^2 + 5*n + 41, 25*n^2 + 15*n + 43, ..., 25*n^2 + 5*(2*k+1)*n + p, ..., 25*n^2 + 5*79*n + 1601, where p is a (prime) term of the Euler polynomial p = k^2 + k + 41, from k=0 to k=39, have their discriminant equal to -4075 = -163*5^2.
%H A181963 Bruno Berselli, <a href="/A181963/b181963.txt">Table of n, a(n) for n = 0..1000</a>
%H A181963 Factor Database, <a href="http://www.factorization.ath.cx/index.php?query=25*n%5E2-1185*n%2B14083&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=50&format=1">Factorizations of 25n^2-1185n+14083</a>.
%H A181963 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A181963 G.f.: (14083 - 29326*x + 15293*x^2)/(1-x)^3. - _Bruno Berselli_, Apr 06 2012
%F A181963 From _Elmo R. Oliveira_, Feb 09 2025: (Start)
%F A181963 E.g.f.: exp(x)*(14083 - 1160*x + 25*x^2).
%F A181963 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A181963 Table[25*n^2 - 1185*n + 14083, {n, 0, 50}] (* _T. D. Noe_, Apr 04 2012 *)
%t A181963 LinearRecurrence[{3,-3,1},{14083,12923,11813},50] (* _Harvey P. Dale_, Aug 28 2022 *)
%o A181963 (Magma) [n^2-237*n+14083: n in [0..220 by 5]]; // _Bruno Berselli_, Apr 06 2012
%o A181963 (PARI) a(n)=25*n^2-1185*n+14083 \\ _Charles R Greathouse IV_, Jun 17 2017
%K A181963 nonn,easy
%O A181963 0,1
%A A181963 _Marius Coman_, Apr 04 2012
%E A181963 Offset changed from 1 to 0 by _Bruno Berselli_, Apr 06 2012