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A322830 a(n) = 32*n^3 + 48*n^2 + 18*n + 1.

%I A322830 #36 Mar 11 2019 11:34:14
%S A322830 1,99,485,1351,2889,5291,8749,13455,19601,27379,36981,48599,62425,
%T A322830 78651,97469,119071,143649,171395,202501,237159,275561,317899,364365,
%U A322830 415151,470449,530451,595349,665335,740601,821339,907741,999999,1098305,1202851,1313829,1431431,1555849
%N A322830 a(n) = 32*n^3 + 48*n^2 + 18*n + 1.
%H A322830 Seiichi Manyama, <a href="/A322830/b322830.txt">Table of n, a(n) for n = 0..10000</a>
%H A322830 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A322830 a(n) + sqrt(a(n)^2 - 1) = (sqrt(n+1) + sqrt(n))^6.
%F A322830 a(n) - sqrt(a(n)^2 - 1) = (sqrt(n+1) - sqrt(n))^6.
%F A322830 a(n) = A144129(2*n+1) = T_3(2*n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind.
%F A322830 From _Colin Barker_, Dec 27 2018: (Start)
%F A322830 G.f.: (1 + x)*(1 + 94*x + x^2) / (1 - x)^4.
%F A322830 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
%F A322830 a(n) = (2*n + 1)*(16*n^2 + 16*n + 1). - _Bruno Berselli_, Jan 02 2019
%e A322830 (sqrt(2) + sqrt(1))^6 = 99 + 70*sqrt(2) = 99 + sqrt(99^2 - 1). So a(1) = 99.
%p A322830 [32*n^3+48*n^2+18*n+1$n=0..40]; # _Muniru A Asiru_, Jan 02 2019
%t A322830 CoefficientList[Series[(1 + x) (1 + 94 x + x^2)/(1 - x)^4, {x, 0, 36}], x] (* or *)
%t A322830 Array[ChebyshevT[3, 2 # + 1] &, 37, 0] (* _Michael De Vlieger_, Jan 01 2019 *)
%t A322830 Table[32n^3+48n^2+18n+1,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,99,485,1351},40] (* _Harvey P. Dale_, Mar 11 2019 *)
%o A322830 (PARI) {a(n) = 32*n^3+48*n^2+18*n+1}
%o A322830 (PARI) {a(n) = polchebyshev(3, 1, 2*n+1)}
%o A322830 (PARI) Vec((1 + x)*(1 + 94*x + x^2) / (1 - x)^4 + O(x^40)) \\ _Colin Barker_, Dec 27 2018
%o A322830 (GAP) a:=List([0..40],n->32*n^3+48*n^2+18*n+1);; Print(a); # _Muniru A Asiru_, Jan 02 2019
%Y A322830 Column 3 of A322790.
%Y A322830 Cf. A144129.
%K A322830 nonn,easy
%O A322830 0,2
%A A322830 _Seiichi Manyama_, Dec 27 2018