A138513 - OEIS

%I #18 Sep 26 2024 23:15:46

%S 1,3,19,137,1001,7323,53579,392017,2868241,20985843,153545539,

%T 1123435097,8219753081,60140849163,440028027899,3219519977377,

%U 23556019679521,172350557549283,1261024361996659,9226442108226857

%N a(n) = 8*a(n-1) - 5*a(n-2).

%C Rightmost digit of each term forms a cycle with period 4: 1, 3, 9, 7, ... (repeat) ...

%C Limit_{n->oo} a(n)/a(n-1) = 4 + sqrt(11) = 7.31662479...

%H Vincenzo Librandi, <a href="/A138513/b138513.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-5).

%F a(n) = 8*a(n-1) - 5*a(n-2), n> 2; given a(1) = 1, a(2) = 3.

%F a(n) = upper left term of the 2 X 2 matrix [1,2; 1,7]^n * [1,0].

%F O.g.f.: x*(1-5*x)/(1-8*x+5*x^2). - _R. J. Mathar_, Apr 12 2008

%F a(n) = (3*sqrt(11)/22 + 1/2)*(4 - sqrt(11))^n + (-3*sqrt(11)/22 + 1/2)*(4 + sqrt(11))^n. - _Emeric Deutsch_, Apr 02 2008

%e a(5) = 1001 = 8*a(4) - 5*a(3) = 8*137 - 5*19.

%e a(5) = 1001 = upper left term in [1,2; 1,7]^5.

%p a[1]:=1: a[2]:=3: for n from 3 to 25 do a[n]:=8*a[n-1]-5*a[n-2] end do: seq(a[n],n=1..20); # _Emeric Deutsch_, Apr 02 2008

%t LinearRecurrence[{8,-5}, {1,3}, 50] (* _G. C. Greubel_, Sep 28 2017 *)

%o (PARI) x='x+O('x^50); Vec(x*(1-5*x)/(1-8*x+5*x^2)) \\ _G. C. Greubel_, Sep 28 2017

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Mar 22 2008

%E More terms from _R. J. Mathar_ and _Emeric Deutsch_, Apr 12 2008