A296488 Decimal expansion of limiting power-ratio for A293076; see Comments.

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

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 = A293076, 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 = 4.863698868156079195859888752149657198717...

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];
j = 1; While[j < 13, 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}]; (* A293076 *)
z = 2000; 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]   (* A296488 *)

Crossrefs

Cf. A001622, A293076, A296284, A296487.

Sequence in context: A201335 A338942 A219246 * A199294 A155741 A243376

Adjacent sequences: A296485 A296486 A296487 * A296489 A296490 A296491

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 19 2017

Status

approved