|
%I #22 Mar 28 2023 07:31:33
%S 14,16,22,32,46,64,86,112,142,176,214,256,302,352,406,464,526,592,662,
%T 736,814,896,982,1072,1166,1264,1366,1472,1582,1696,1814,1936,2062,
%U 2192,2326,2464,2606,2752,2902,3056,3214,3376,3542,3712,3886,4064,4246,4432
%N a(n) = 2*n^2 + 14.
%C This is the case k=7 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.
%C Equivalently, numbers m such that 2*m - 28 is a square.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: 2*(7 - 13*x + 8*x^2)/(1 - x)^3.
%F a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = 2*A117619(n).
%F From _Amiram Eldar_, Mar 28 2023: (Start)
%F Sum_{n>=0} 1/a(n) = (1 + sqrt(7)*Pi*coth(sqrt(7)*Pi))/28.
%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(7)*Pi*cosech(sqrt(7)*Pi))/28. (End)
%t Table[2 n^2 + 14, {n, 0, 50}]
%o (PARI) vector(50, n, n--; 2*n^2+14)
%o (Sage) [2*n^2+14 for n in (0..50)]
%o (Magma) [2*n^2+14: n in [0..50]];
%Y Cf. A117619.
%Y Subsequence of A047235 and A047451.
%Y Cf. similar sequences listed in A255843.
%K nonn,easy
%O 0,1
%A _Avi Friedlich_, Mar 08 2015
%E Edited by _Bruno Berselli_, Mar 13 2015
|
