%I #8 Jan 07 2018 10:51:22

%S 3,6,8,4,1,4,0,9,6,3,5,0,7,1,4,3,0,4,8,9,3,4,9,9,8,7,0,9,4,3,3,9,8,3,

%T 2,1,8,5,0,7,1,3,4,7,6,0,3,4,5,1,0,3,4,9,8,0,8,8,6,8,7,6,5,9,1,0,0,0,

%U 8,3,6,4,2,0,6,9,0,2,9,5,4,8,8,5,7,4

%N Decimal expansion of limiting power-ratio for A294557; see Comments.

%C Suppose that A = (a(n)), for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A294557, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e limiting power-ratio = 3.684140963507143048934998709433983218507...

%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - n;

%t j = 1; While[j < 13, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

%t Table[a[n], {n, 0, k}]; (* A294557 *)

%t z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];

%t StringJoin[StringTake[ToString[h[[z]]], 41], "..."]

%t Take[RealDigits[Last[h], 10][[1]], 120] (* A296568 *)

%Y Cf. A001622, A294557, A296469, A296567.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 20 2017