A161939 - OEIS

%I #19 Sep 08 2022 08:45:45

%S 3,14,70,364,1932,10360,55832,301616,1631280,8827616,47783008,

%T 258677440,1400457408,7582175104,41050997120,222257525504,

%U 1203346244352,6515164597760,35274469361152,190983450520576,1034025033108480

%N a(n) = ((3+sqrt(2))*(4+sqrt(2))^n + (3-sqrt(2))*(4-sqrt(2))^n)/2.

%C Fourth binomial transform of A162255.

%H Muniru A Asiru, <a href="/A161939/b161939.txt">Table of n, a(n) for n = 0..280</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) = 3; a(1) = 14.

%F G.f.: (3-10*x)/(1-8*x+14*x^2).

%p seq(simplify(((3+sqrt(2))*(4+sqrt(2))^n+(3-sqrt(2))*(4-sqrt(2))^n)*1/2), n = 0 .. 20); # _Emeric Deutsch_, Jun 28 2009

%t LinearRecurrence[{8,-14},{3,14},30] (* _Harvey P. Dale_, May 10 2012 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 01 2009

%o (GAP) a := [3, 14];; for n in [3..10^2] do a[n] := 8*a[n-1] - 14*a[n-2]; od; a; # _Muniru A Asiru_, Feb 02 2018

%o (PARI) x='x+O('x^30); Vec((3-10*x)/(1-8*x+14*x^2)) \\ _G. C. Greubel_, Aug 17 2018

%Y Cf. A162255, A161940 (Fifth binomial transform of A162255).

%K nonn

%O 0,1

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

%E Definition corrected by _Emeric Deutsch_, Jun 28 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 01 2009

%E Extended by _Emeric Deutsch_, Jun 28 2009