A164360 - OEIS

%I #20 Dec 12 2023 08:09:41

%S 5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,

%T 4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,

%U 3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3,5,4,3

%N Period 3: repeat [5, 4, 3].

%C From _Klaus Brockhaus_, May 29 2010: (Start)

%C Continued fraction expansion of (32+sqrt(1297))/13.

%C Decimal expansion of 181/333. (End)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).

%F a(n) = 4+(-1)^n*((1/2+I*sqrt(3)/6)*((1+I*sqrt(3))/2)^n+(1/2-I*sqrt(3)/6)*((1-I*sqrt(3))/2)^n). [Corrected by _Klaus Brockhaus_, Sep 17 2009]

%F a(n) = 4+(1/3)*sqrt(3)*sin(2*n*Pi/3)+cos(2*n*Pi/3). [Corrected by _Klaus Brockhaus_, Sep 17 2009]

%F a(n) = a(n-3) for n > 2, with a(0) = 5, a(1) = 4, a(2) = 3.

%F G.f.: (5+4*x+3*x^2)/((1-x)*(1+x+x^2)). [_Klaus Brockhaus_, Sep 17 2009]

%F E.g.f.: 4*exp(x)+(1/3)*sqrt(3)*exp(-(1/2)*x)*sin((1/2)*x*sqrt(3))+exp(-(1/2)*x)*cos((1/2)*x*sqrt(3)).

%F a(n) = 4 + A057078(n). - _Wesley Ivan Hurt_, Jul 01 2016

%p seq(op([5, 4, 3]), n=0..50); # _Wesley Ivan Hurt_, Jul 01 2016

%t PadRight[{}, 100, {5, 4, 3}] (* _Wesley Ivan Hurt_, Jul 01 2016 *)

%o (Magma) [ n le 3 select 6-n else Self(n-3):n in [1..105] ]; // _Klaus Brockhaus_, Sep 17 2009

%o (Magma) &cat [[5, 4, 3]^^30]; // _Wesley Ivan Hurt_, Jul 01 2016

%Y Cf. A007877 (repeat 0,1,2,1), A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2), A158289 (repeat 0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1).

%Y Cf. A178566 (decimal expansion of (32+sqrt(1297))/13). [_Klaus Brockhaus_, May 29 2010]

%K easy,nonn

%O 0,1

%A _Stephen Crowley_, Aug 14 2009

%E Edited by _Klaus Brockhaus_, Sep 17 2009

%E Offset changed to 0 and formulas adjusted by _Klaus Brockhaus_, May 18 2010