A229277 - OEIS

%I #18 Jul 01 2023 15:57:54

%S 0,3,15,63,243,891,3159,10935,37179,124659,413343,1358127,4428675,

%T 14348907,46235367,148272039,473513931,1506635235,4778186031,

%U 15109399071,47652720147,149931729243,470715894135,1474909801623,4613015762523,14403906360531,44906296300479

%N Number of ascending runs in {1,...,3}^n.

%H Alois P. Heinz, <a href="/A229277/b229277.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 (6,-9).

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

%F a(n) = 3^(n-1)*(2*n+1) for n>0, a(0) = 0.

%F a(n) = 3*A081038(n-1) for n>0.

%F From _Amiram Eldar_, May 17 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 3*(sqrt(3)*arctanh(1/sqrt(3)) - 1).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - sqrt(3)*Pi/2. (End)

%p a:= n-> `if`(n=0, 0, 3^(n-1)*(2*n+1)):

%p seq(a(n), n=0..30);

%t a[0] = 0; a[n_] := 3^(n - 1)*(2*n + 1); Array[a, 30, 0] (* _Amiram Eldar_, May 17 2022 *)

%Y Column k=3 of A229079.

%Y Cf. A081038.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Sep 18 2013