A296845 Decimal expansion of limiting power-ratio for A296843; see Comments.

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

Offset

1,1

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 = A296843, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Links

Table of n, a(n) for n=1..86.

Example

limiting power-ratio = 6.136385518666220790955343949152636124460...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];
j = 1; While[j < 16, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}];  (* A296843 *)
z = 1700; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120] (* A296845 *)

Crossrefs

Cf. A001622, A296469, A296843, A296844.

Sequence in context: A249282 A289090 A260691 * A030644 A319032 A344125

Adjacent sequences: A296842 A296843 A296844 * A296846 A296847 A296848

Keyword

nonn,easy,cons

Author

Clark Kimberling, Jan 12 2018

Status

approved