%I #4 Dec 18 2017 19:46:18

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

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

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

%N Decimal expansion of limiting power-ratio for A296288; 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 = A296288 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

%e limiting power-ratio = 19.20919895224276906694327100124445652349...

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

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

%t j = 1; While[j < 12, 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, 15}] (* A296288 *)

%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] (* A296460 *)

%Y Cf. A001622, A296288.

%K nonn,easy,cons

%O 2,2

%A _Clark Kimberling_, Dec 18 2017