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a(n) = 2*n^2 + 16*n + 6.
1

%I #25 Mar 02 2023 02:53:24

%S 24,46,72,102,136,174,216,262,312,366,424,486,552,622,696,774,856,942,

%T 1032,1126,1224,1326,1432,1542,1656,1774,1896,2022,2152,2286,2424,

%U 2566,2712,2862,3016,3174,3336,3502,3672,3846,4024,4206,4392,4582,4776,4974,5176

%N a(n) = 2*n^2 + 16*n + 6.

%C Eighth diagonal of A144562.

%C 2*a(n) + 52 is a square.

%H Harvey P. Dale, <a href="/A154590/b154590.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 2*A116711(n+3).

%F G.f.: -2*x*(3*x-4)*(x-3)/(x-1)^3.

%F From _Amiram Eldar_, Mar 02 2023: (Start)

%F Sum_{n>=1} 1/a(n) = 35/468 - cot(sqrt(13)*Pi)*Pi/(4*sqrt(13)).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 121/468 + cosec(sqrt(13)*Pi)*Pi/(4*sqrt(13)). (End)

%t Table[2n^2+16n+6,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{24,46,72},50] (* _Harvey P. Dale_, Dec 27 2011 *)

%o (PARI) a(n)=2*n^2+16*n+6 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A144562, A067076, A116711, A153238.

%K nonn,easy,less

%O 1,1

%A _Vincenzo Librandi_, Jan 12 2009

%E Corrected (a(31) added) by _Harvey P. Dale_, Dec 27 2011