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a(n) = n^2 + 10.
4

%I #34 Feb 12 2024 02:28:50

%S 11,14,19,26,35,46,59,74,91,110,131,154,179,206,235,266,299,334,371,

%T 410,451,494,539,586,635,686,739,794,851,910,971,1034,1099,1166,1235,

%U 1306,1379,1454,1531,1610,1691,1774,1859,1946,2035,2126,2219,2314,2411,2510

%N a(n) = n^2 + 10.

%C Conjecture: n^2 + 10 != 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 _Amiram Eldar_, Nov 02 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + sqrt(10)*Pi*coth(sqrt(10)*Pi))/20.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(10)*Pi*cosech(sqrt(10)*Pi))/20. (End)

%F From _Amiram Eldar_, Feb 12 2024: (Start)

%F Product_{n>=0} (1 - 1/a(n)) = (3/sqrt(10))*sinh(3*Pi)/sinh(sqrt(10)*Pi).

%F Product_{n>=0} (1 + 1/a(n)) = sqrt(11/10)*sinh(sqrt(11)*Pi)/sinh(sqrt(10)*Pi). (End)

%t a[n_]:=n^2+10; a[Range[200]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2011*)

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Feb 21 2006

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