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A163827 a(n) = 6n^3 + 1, solution z in Diophantine equation x^3 + y^3 = z^3 - 2. It may be considered a Fermat near miss by 2.

%I A163827 #18 Sep 08 2022 08:45:47
%S A163827 7,49,163,385,751,1297,2059,3073,4375,6001,7987,10369,13183,16465,
%T A163827 20251,24577,29479,34993,41155,48001,55567,63889,73003,82945,93751,
%U A163827 105457,118099,131713,146335,162001,178747,196609,215623,235825,257251,279937
%N A163827 a(n) = 6n^3 + 1, solution z in Diophantine equation x^3 + y^3 = z^3 - 2. It may be considered a Fermat near miss by 2.
%C A163827 It is easy to check that with x = 6n^2, y = 6n^3 - 1, and this z = 6n^3 + 1, it satisfies the Diophantine equation x^3 + y^3 = z^3 - 2. Thus these are near-misses for Fermat equation.
%C A163827 For n>2, it seems to be the only solution of x^n + y^n = z^n - 2 (or even that differ by 2 from FLT, see A050787 and A050791 for solutions that differ by 1). As 2 is not a cube, these solutions are not included in the theory for x^3 + y^3 = u^3 + v^3.
%H A163827 Harvey P. Dale, <a href="/A163827/b163827.txt">Table of n, a(n) for n = 1..1000</a>
%H A163827 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F A163827 a(n) = 6n^3+1.
%F A163827 a(1)=7, a(2)=49, a(3)=163, a(4)=385, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). [_Harvey P. Dale_, Dec 12 2011]
%F A163827 G.f.: (-x^3+9*x^2+21*x+7)/(x-1)^4. [_Harvey P. Dale_, Dec 12 2011]
%e A163827 For n=1, a(1)=7 and 7^3 - 2 (=341) = 5^3 + 6^3.
%e A163827 For n=2, a(2)=49 and 49^3 - 2 (=117647) = 24^3 + 47^3.
%p A163827 A163827:=n->6*n^3+1: seq(A163827(n), n=1..50); # _Wesley Ivan Hurt_, Jan 09 2017
%t A163827 6*Range[40]^3+1 (* or *) LinearRecurrence[{4,-6,4,-1},{7,49,163,385},40] (* _Harvey P. Dale_, Dec 12 2011 *)
%o A163827 (PARI) a(n)=6*n^3+1 \\ _Charles R Greathouse IV_, Nov 29 2014
%o A163827 (Magma) [6*n^3+1 : n in [1..50]]; // _Wesley Ivan Hurt_, Jan 09 2017
%Y A163827 Cf. A050787, A050791.
%K A163827 nonn,easy
%O A163827 1,1
%A A163827 _Carlos Alves_, Aug 05 2009