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a(n) = n^2 + 34.
1

%I #22 Jan 26 2025 09:12:48

%S 34,35,38,43,50,59,70,83,98,115,134,155,178,203,230,259,290,323,358,

%T 395,434,475,518,563,610,659,710,763,818,875,934,995,1058,1123,1190,

%U 1259,1330,1403,1478,1555,1634,1715,1798,1883,1970,2059,2150,2243,2338,2435

%N a(n) = n^2 + 34.

%C Conjecture: n^2 + 34 != x^k for all n,x and k > 1.

%C The conjecture is true: See Cohn. - _James Rayman_, Feb 14 2023

%H J. H. E. Cohn, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa65/aa6546.pdf">The diophantine equation x^2 + C = y^n</a>, Acta Arithmetica LXV.4 (1993).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Elmo R. Oliveira_, Jan 25 2025: (Start)

%F G.f.: (34 - 67*x + 35*x^2)/(1 - x)^3.

%F E.g.f.: (34 + x + x^2)*exp(x).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

%t 34+Range[50]^2 (* _Harvey P. Dale_, Jan 28 2011 *)

%o (PARI) a(n)=n^2+34

%Y Cf. A114948, A114962, A114963, A114964, A241850.

%K easy,nonn,changed

%O 0,1

%A _Cino Hilliard_, Feb 21 2006

%E Edited by _Charles R Greathouse IV_, Aug 09 2010

%E a(0) = 34 prepended by _Elmo R. Oliveira_, Jan 26 2025