%I #18 Sep 08 2022 08:45:45

%S 2,5,16,54,184,628,2144,7320,24992,85328,291328,994656,3395968,

%T 11594560,39586304,135156096,461451776,1575494912,5379076096,

%U 18365314560,62703106048,214081795072,730920968192,2495520282624,8520239194112

%N a(n) = ((4+sqrt(2))*(2+sqrt(2))^n + (4-sqrt(2))*(2-sqrt(2))^n)/4.

%C Second binomial transform of A135530.

%H G. C. Greubel, <a href="/A161941/b161941.txt">Table of n, a(n) for n = 0..1000</a>

%H C. Bautista-Ramos and C. Guillen-Galvan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Bautista/bautista4.html">Fibonacci numbers of generalized Zykov sums</a>, J. Integer Seq., 15 (2012), #12.7.8.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2).

%F a(n) = 4*a(n-1) - 2*a(n-2) for n>1; a(0) = 2; a(1) = 5.

%F G.f.: (2-3*x)/(1-4*x+2*x^2).

%F a(n) = 2*A007070(n) - 3*A007070(n-1). - _R. J. Mathar_, Oct 20 2017

%t LinearRecurrence[{4,-2},{2,5},30] (* _Harvey P. Dale_, May 26 2012 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((4+r)*(2+r)^n+(4-r)*(2-r)^n)/4: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 01 2009

%o (PARI) x='x+O('x^30); Vec((2-3*x)/(1-4*x+2*x^2)) \\ _G. C. Greubel_, Jan 27 2018

%Y Cf. A135530, A161944 (third binomial transform of A135530).

%K nonn,easy

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

%E Edited and extended beyond a(4) by _Klaus Brockhaus_, Jul 01 2009