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A255842 a(n) = 2*n^2 + 12.

%I A255842 #51 Jan 25 2025 10:49:29
%S A255842 12,14,20,30,44,62,84,110,140,174,212,254,300,350,404,462,524,590,660,
%T A255842 734,812,894,980,1070,1164,1262,1364,1470,1580,1694,1812,1934,2060,
%U A255842 2190,2324,2462,2604,2750,2900,3054,3212,3374,3540,3710,3884,4062,4244,4430
%N A255842 a(n) = 2*n^2 + 12.
%C A255842 This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.
%C A255842 Equivalently, numbers m such that 2*m - 24 is a square.
%C A255842 For n = 0..10, a(n) - 1 is prime (see A092968).
%H A255842 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A255842 G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.
%F A255842 a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A255842 a(n) = 2*A114949(n).
%F A255842 From _Amiram Eldar_, Mar 28 2023: (Start)
%F A255842 Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.
%F A255842 Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)
%F A255842 E.g.f.: 2*exp(x)*(6 + x + x^2). - _Elmo R. Oliveira_, Jan 24 2025
%t A255842 Table[2 n^2 + 12, {n, 0, 50}]
%o A255842 (PARI) vector(50, n, n--; 2*n^2+12)
%o A255842 (Sage) [2*n^2+12 for n in (0..50)]
%o A255842 (Magma) [2*n^2+12: n in [0..50]];
%Y A255842 Cf. A016825 (first differences), A092968, A114949.
%Y A255842 Subsequence of A047238 and A047406.
%Y A255842 Cf. similar sequences listed in A255843.
%K A255842 nonn,easy
%O A255842 0,1
%A A255842 _Avi Friedlich_, Mar 08 2015
%E A255842 Edited by _Bruno Berselli_, Mar 11 2015