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%I #71 Sep 08 2022 08:46:01
%S 41,43,47,53,61,71,83,97,113,131,151,173,197,223,251,281,313,347,383,
%T 421,461,503,547,593,641,691,743,797,853,911,971,1033,1097,1163,1231,
%U 1301,1373,1447,1523,1601,1681,1763,1847,1933,2021,2111,2203,2297,2393
%N a(n) = n^2 + n + 41.
%C Euler's famous prime-generating polynomial; a(0) through a(39) are all prime.
%H Reinhard Zumkeller, <a href="/A202018/b202018.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A005846(n) for n < 41, a(41) = A145292(1);
%F Union of A005846 (primes) and A145292 (composites);
%F a(n) = A002378(n) + 41.
%F a(a(n) + n) = a(n)*a(n+1). - _Vladimir Shevelev_, Jul 16 2012 (This identity holds for all sequences of the form n^2 + n + c, _Joerg Arndt_, Jul 17 2012).
%F a(0) = 41 and for n > 0, a(n) = a(n-1) + 2*n. - _Jean-Christophe Hervé_, Sep 27 2014
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Colin Barker_, Sep 28 2014
%F G.f.: (41*x^2 - 80*x + 41) / (1-x)^3. - _Colin Barker_, Sep 28 2014
%F a(n) = 2*a(n-1) - a(n-2) + 2. - _Vincenzo Librandi_, Mar 04 2016
%F E.g.f.: (x^2 + 2*x + 41)*exp(x). - _Robert Israel_, Mar 10 2016
%p A202018:=n->n^2+n+41: seq(A202018(n), n=0..50); # _Wesley Ivan Hurt_, Sep 28 2014
%t Table[n^2 + n + 41, {n, 0, 49}] (* _Alonso del Arte_, Dec 08 2011
%o (Haskell)
%o a202018 = (+ 41) . a002378
%o (PARI) a(n)=n^2+n+41 \\ _Charles R Greathouse IV_, Dec 08 2011
%o (Magma) [n^2 + n + 41 : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 28 2014
%o (Scala) (0 to 49).map((n: Int) => n * n + n + 41) // _Alonso del Arte_, Nov 29 2018
%o (Sage) [n^2+n+41 for n in range(50)] # _G. C. Greubel_, Dec 04 2018
%o (GAP) List([0..50], n -> n^2 +n+41); # _G. C. Greubel_, Dec 04 2018
%Y Cf. A060566, A010051, A000040, A002808.
%Y Cf. A002378, A005846, A145292.
%Y Cf. A056561.
%K nonn,easy
%O 0,1
%A _Reinhard Zumkeller_, Dec 08 2011
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