A275155 a(1) = 18; a(n) = 3*a(n - 1) + 2*sqrt(2*a(n - 1)*(a(n - 1) - 14)) - 14 for n > 1.

18, 64, 338, 1936, 11250, 65536, 381938, 2226064, 12974418, 75620416, 440748050, 2568867856, 14972459058, 87265886464, 508622859698, 2964471271696, 17278204770450, 100704757350976, 586950339335378, 3420997278661264, 19939033332632178, 116213202717131776

Offset: 1

Author

Manuel López Holgueras, Jul 18 2016

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x*(9-31*x+8*x^2)/((1-x)*(1-6*x+x^2)))); // G. C. Greubel, Sep 30 2018

Mathematica

NestList[3 # + 2 Sqrt[2 # (# - 14)] - 14 &, 18, 18] (* Michael De Vlieger, Jul 19 2016 *)
CoefficientList[Series[2*x*(9-31*x+8*x^2)/((1-x)*(1-6*x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Sep 30 2018 *)

PARI

a(n)=([0,1,0; 0,0,1; 1,-7,7]^(n-1)*[18;64;338])[1,1] \\ Charles R Greathouse IV, Jul 20 2016

PARI

Vec(2*x*(9-31*x+8*x^2)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Jul 21 2016

Formula

a(n+1) = 3*a(n) + 2*sqrt(2*a(n)*(a(n) - 14)) - 14.

From Colin Barker, Jul 21 2016: (Start)

a(n) = (14+(9-4*sqrt(2))*(3+2*sqrt(2))^n + (3-2*sqrt(2))^n*(9+4*sqrt(2)))/2.

a(n) = 7*a(n-1) -7*a(n-2) +a(n-3) for n>3.

G.f.: 2*x*(9-31*x+8*x^2) / ((1-x)*(1-6*x+x^2)). (End)