|
%I #8 Sep 08 2022 08:45:47
%S 2,11,60,326,1768,9580,51888,280984,1521440,8237744,44601792,
%T 241485920,1307462272,7078895296,38326690560,207508990336,
%U 1123498254848,6082860174080,32933905824768,178311204161024,965414951741440
%N a(n) = ((4+3*sqrt(2))*(4+sqrt(2))^n + (4-3*sqrt(2))*(4-sqrt(2))^n)/4.
%C Binomial transform of A164033. Fourth binomial transform of A164090. Inverse binomial transform of A164035.
%H G. C. Greubel, <a href="/A164034/b164034.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-14).
%F a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 2, a(1) = 11.
%F G.f.: (2-5*x)/(1-8*x+14*x^2).
%F E.g.f.: (2*cosh(sqrt(2)*x) + (3*sqrt(2)/2)*sinh(sqrt(2)*x))*exp(4*x). - _G. C. Greubel_, Sep 08 2017
%t LinearRecurrence[{8,-14},{2,11},30] (* _Harvey P. Dale_, Aug 09 2016 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((4+3*r)*(4+r)^n+(4-3*r)*(4-r)^n)/4: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 09 2009
%o (PARI) x='x+O('x^50); Vec((2-5*x)/(1-8*x+14*x^2)) \\ _G. C. Greubel_, Sep 08 2017
%Y Cf. A164033, A164090, A164035.
%K nonn
%O 0,1
%A Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 09 2009
| 