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A202018
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a(n) = n^2 + n + 41.
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21
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41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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Euler's famous prime-generating polynomial; a(0) through a(39) are all prime.
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LINKS
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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).
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FORMULA
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a(n) = A005846(n) for n < 41, a(41) = A145292(1);
Union of A005846 (primes) and A145292 (composites);
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
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MAPLE
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A202018:=n->n^2+n+41: seq(A202018(n), n=0..50); # Wesley Ivan Hurt, Sep 28 2014
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MATHEMATICA
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Table[n^2 + n + 41, {n, 0, 49}] (* Alonso del Arte, Dec 08 2011
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PROG
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(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
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CROSSREFS
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Cf. A060566, A010051, A000040, A002808.
Cf. A002378, A005846, A145292.
Cf. A056561.
Sequence in context: A330673 A296921 A155884 * A005846 A273756 A154498
Adjacent sequences: A202015 A202016 A202017 * A202019 A202020 A202021
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KEYWORD
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nonn,easy
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AUTHOR
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Reinhard Zumkeller, Dec 08 2011
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STATUS
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approved
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