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%I #10 Mar 11 2017 21:16:37
%S 2,8,40,216,1218,7000,40560,235824,1373090,7999592,46616920,271683720,
%T 1583441442,9228858808,53789455200,313507253856,1827252574658,
%U 10650004589000,62072766255880,361786571934264,2108646614622210
%N Sum of the square root of n-th square triangular number and n-th Pell (or lambda) number (A000129).
%H G. C. Greubel, <a href="/A113449/b113449.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-12,-4,1).
%F a(n) = sqrt(((17 + 12*sqrt(2))^n + (17 - 12*sqrt(2))^n - 2)/32) + ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/(2*sqrt(2)). - _Stefan Steinerberger_, Jun 17 2007
%F From _G. C. Greubel_, Mar 11 2017: (Start)
%F a(n) = sqrt((Q_{4*n} - 2)/32) + P_{n}, where P_{n} and the Pell numbers and Q_{n} are the Pell-Lucas numbers.
%F a(n) = 8*a(n-1) - 12*a(n-2) - 4*a(n-3) + a(n-4).
%F G.f.: (2*x)*(1-4*x) / ((1-2*x-x^2)*(1-6*x+x^2)). (End)
%t Simplify[Table[Sqrt[((17 + 12*Sqrt[2])^n + (17 - 12*Sqrt[2])^n - 2)/32] + ((1 + Sqrt[2])^n - (1 - Sqrt[2])^n)/(2*Sqrt[2]), {n, 1, 25}]] (* _Stefan Steinerberger_, Jun 17 2007 *)
%t Table[Sqrt[(LucasL[4*n, 2] - 2)/32] + Fibonacci[n, 2], {n,1,50}] (* _G. C. Greubel_, Mar 11 2017 *)
%o (PARI) x='x+O('x^50); Vec((2*x)*(1-4*x) / ((1-2*x-x^2)*(1-6*x+x^2))) \\ _G. C. Greubel_, Mar 11 2017
%Y Cf. A000129, A001110, A002203.
%K easy,nonn
%O 1,1
%A K. B. Subramaniam (subramaniam_kb05(AT)yahoo.co.in), Nov 02 2005
%E More terms from _Stefan Steinerberger_, Jun 17 2007
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