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a(n) = n^2 + 23.
2

%I #28 Nov 29 2024 19:08:23

%S 23,24,27,32,39,48,59,72,87,104,123,144,167,192,219,248,279,312,347,

%T 384,423,464,507,552,599,648,699,752,807,864,923,984,1047,1112,1179,

%U 1248,1319,1392,1467,1544,1623,1704,1787,1872,1959,2048,2139,2232,2327,2424,2523

%N a(n) = n^2 + 23.

%H Vincenzo Librandi, <a href="/A241889/b241889.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: (23 - 45*x + 24*x^2)/(1 - x)^3.

%F a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.

%F From _Amiram Eldar_, Nov 04 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + sqrt(23)*Pi*coth(sqrt(23)*Pi))/46.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(23)*Pi*cosech(sqrt(23)*Pi))/46. (End)

%F E.g.f.: exp(x)*(23 + x + x^2). - _Elmo R. Oliveira_, Nov 29 2024

%t CoefficientList[Series[(23 - 45 x + 24 x^2)/(1 - x)^3,{x, 0, 60}], x]

%t Range[0, 50]^2 + 23 (* or *) LinearRecurrence[{3, -3, 1}, {23, 24, 27}, 50] (* _Harvey P. Dale_, May 27 2014 *)

%o (Magma) [n^2+23: n in [0..60]];

%o (PARI) a(n)=n^2+23 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. similar sequences listed in A114962.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, May 01 2014