A082106 - OEIS

%I #18 Jan 08 2023 02:42:09

%S 1,6,33,118,321,726,1441,2598,4353,6886,10401,15126,21313,29238,39201,

%T 51526,66561,84678,106273,131766,161601,196246,236193,281958,334081,

%U 393126,459681,534358,617793,710646,813601,927366,1052673,1190278,1340961,1505526,1684801

%N Main diagonal of number array A082105.

%C 4*a(n) can be written as (n^2 + 2*n + 1)^2 + (n^2 - 2*n + 1)^2 + (n^2 - 2*n - 1)^2 + (n^2 + 2*n - 1)^2. - _Bruno Berselli_, Jun 20 2014

%H G. C. Greubel, <a href="/A082106/b082106.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n^4 + 4*n^2 + 1.

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Dec 06 2012

%F G.f.: (1 + x + 13*x^2 + 3*x^3 + 6*x^4)/(1 - x)^5. - _Bruno Berselli_, Jun 20 2014

%F E.g.f.: (1 + 5*x + 11*x^2 + 6*x^3 + x^4)*exp(x). - _G. C. Greubel_, Dec 22 2022

%F Sum_{n>=0} 1/a(n) = 1/2 + (Pi/4)*((1/sqrt(2)+1/sqrt(6))*coth(sqrt(2-sqrt(3))*Pi) - (1/sqrt(2)-1/sqrt(6))*coth(sqrt(2+sqrt(3))*Pi)). - _Amiram Eldar_, Jan 08 2023

%t Table[n^4+4n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,6,33,118,321},40] (* _Harvey P. Dale_, Dec 06 2012 *)

%o (Magma) [(n^2+2)^2 -3: n in [0..40]]; // _G. C. Greubel_, Dec 22 2022

%o (SageMath) [(n^2+2)^2 -3 for n in range(41)] # _G. C. Greubel_, Dec 22 2022

%Y Cf. A082047, A082044, A082105, A082107, A082109.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Apr 03 2003