A297013 Decimal expansion of limiting power-ratio for A297011; see Comments.

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

Offset: 0

Author

Clark Kimberling, Jan 13 2018

Keywords

• nonn
• easy
• cons

Comments

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 = A297011, we have g = 1+ sqrt(2). See the guide at A296469 for related sequences.

Examples

			limiting power-ratio = 0.919401986429269675831327000461742596887...
		

Crossrefs

Cf. A297011.

Programs

Mathematica

a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n];
j = 1; While[j < 9, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A297011 *)
z = 1700; r = 1 + Sqrt[2]; h = Table[N[a[n]/r^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120] (* A297013 *)