A296498 Decimal expansion of limiting power-ratio for A294541; see Comments.

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

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 = A294541, 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 = 5.416437430138486313848494950880888008486...

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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}]; (* A294541 *)
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]   (* A296498 *)

Crossrefs

Cf. A001622, A294541, A296284, A296497.

Sequence in context: A342797 A084129 A011503 * A195297 A336284 A258639

Adjacent sequences: A296495 A296496 A296497 * A296499 A296500 A296501

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 20 2017

Status

approved