OFFSET
0,1
COMMENTS
Euler's famous prime-generating polynomial; a(0) through a(39) are all prime.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = A002378(n) + 41.
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).
a(0) = 41 and for n > 0, a(n) = a(n-1) + 2*n. - Jean-Christophe Hervé, Sep 27 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Colin Barker, Sep 28 2014
G.f.: (41*x^2 - 80*x + 41) / (1-x)^3. - Colin Barker, Sep 28 2014
a(n) = 2*a(n-1) - a(n-2) + 2. - Vincenzo Librandi, Mar 04 2016
E.g.f.: (x^2 + 2*x + 41)*exp(x). - Robert Israel, Mar 10 2016
MAPLE
MATHEMATICA
Table[n^2 + n + 41, {n, 0, 49}] (* Alonso del Arte, Dec 08 2011
PROG
(Haskell)
a202018 = (+ 41) . a002378
(PARI) a(n)=n^2+n+41 \\ Charles R Greathouse IV, Dec 08 2011
(Magma) [n^2 + n + 41 : n in [0..50]]; // Wesley Ivan Hurt, Sep 28 2014
(Scala) (0 to 49).map((n: Int) => n * n + n + 41) // Alonso del Arte, Nov 29 2018
(Sage) [n^2+n+41 for n in range(50)] # G. C. Greubel, Dec 04 2018
(GAP) List([0..50], n -> n^2 +n+41); # G. C. Greubel, Dec 04 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Dec 08 2011
STATUS
approved