This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255844 #48 Jan 25 2025 10:56:36
%S A255844 6,8,14,24,38,56,78,104,134,168,206,248,294,344,398,456,518,584,654,
%T A255844 728,806,888,974,1064,1158,1256,1358,1464,1574,1688,1806,1928,2054,
%U A255844 2184,2318,2456,2598,2744,2894,3048,3206,3368,3534,3704,3878,4056,4238,4424,4614
%N A255844 a(n) = 2*n^2 + 6.
%C A255844 This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.
%C A255844 Equivalently, numbers m such that 2*m-12 is a square.
%C A255844 For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).
%H A255844 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A255844 G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.
%F A255844 a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A255844 a(n) = 2*A117950(n).
%F A255844 From _Amiram Eldar_, Mar 28 2023: (Start)
%F A255844 Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.
%F A255844 Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)
%F A255844 E.g.f.: 2*exp(x)*(3 + x + x^2). - _Elmo R. Oliveira_, Jan 25 2025
%t A255844 Table[2 n^2 + 6, {n, 0, 50}]
%o A255844 (PARI) vector(50, n, n--; 2*n^2+6)
%o A255844 (Sage) [2*n^2+6 for n in (0..50)]
%o A255844 (Magma) [2*n^2+6: n in [0..50]];
%Y A255844 Cf. A016825 (first differences), A087370, A107303, A114949, A117950.
%Y A255844 Cf. A152811: nonnegative numbers of the form 2*m^2-6.
%Y A255844 Subsequence of A000378.
%Y A255844 Cf. similar sequences listed in A255843.
%K A255844 nonn,easy
%O A255844 0,1
%A A255844 _Avi Friedlich_, Mar 08 2015
%E A255844 Corrected and extended by _Bruno Berselli_, Mar 11 2015