%I #4 Dec 18 2017 19:45:57

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

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

%U 5,1,8,0,8,3,2,6,4,8,9,8,3,8,2,4,4,0

%N Decimal expansion of limiting power-ratio for A295862; 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 = A295862 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%F limiting power-ratio = 6.514710075055272773358868575343969425373...

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

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

%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}] (* A295862 *)

%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] (* A296470 *)

%Y Cf. A001622, A295862, A296469.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 18 2017