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A114964 a(n) = n^2 + 30.

%I A114964 #53 Dec 30 2024 13:17:16
%S A114964 30,31,34,39,46,55,66,79,94,111,130,151,174,199,226,255,286,319,354,
%T A114964 391,430,471,514,559,606,655,706,759,814,871,930,991,1054,1119,1186,
%U A114964 1255,1326,1399,1474,1551,1630,1711,1794,1879,1966,2055,2146,2239,2334,2431,2530
%N A114964 a(n) = n^2 + 30.
%C A114964 x^2 + 30 != y^n for all x,y and n > 1, so this is a subsequence of A007916.
%C A114964 From _Bruno Berselli_, May 12 2014: (Start)
%C A114964 This is the case k=5 of the identity n^2 + k*(k+1) = (Sum_{i=-k..k} (n+i)^3)/((2*k+1)*n).
%C A114964 Similar sequences: A059100 (k=1), A114949 (k=2), A241748 (k=3), A241850 (k=4). (End)
%C A114964 The old name of this sequence was: Numbers of the form x^2 + 30. Also numbers that are not a perfect power.
%H A114964 Shawn A. Broyles, <a href="/A114964/b114964.txt">Table of n, a(n) for n = 0..1000</a>
%H A114964 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 A114964 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A114964 From _Amiram Eldar_, Nov 04 2020: (Start)
%F A114964 Sum_{n>=0} 1/a(n) = (1 + sqrt(30)*Pi*coth(sqrt(30)*Pi))/60.
%F A114964 Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(30)*Pi*cosech(sqrt(30)*Pi))/60. (End)
%F A114964 From _Elmo R. Oliveira_, Dec 30 2024: (Start)
%F A114964 G.f.: (30 - 59*x + 31*x^2)/(1 - x)^3.
%F A114964 E.g.f.: (30 + x + x^2)*exp(x).
%F A114964 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)
%e A114964 11*4*a(4) = (-1)^3 + 0^3 + 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = 2024. - _Bruno Berselli_, May 12 2014
%t A114964 Range[0,60]^2+30 (* _Harvey P. Dale_, Oct 17 2022 *)
%o A114964 (PARI) g(n,p) = for(x=0,n,y=x^2+p;print1(y","));
%o A114964 (PARI) a(n) = n^2 + 30; \\ _Altug Alkan_, Apr 30 2018
%Y A114964 Cf. A007916, A059100, A114949, A241748, A241850.
%K A114964 nonn,easy
%O A114964 0,1
%A A114964 _Cino Hilliard_, Feb 21 2006
%E A114964 New name from _Shawn A. Broyles_ and _Altug Alkan_, Apr 30 2018