A296566 Decimal expansion of limiting power-ratio for A294556; see Comments.

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

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 = A294556, 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 = 9.065450226732332928989975992626515946110...

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 - 1] + b[n - 2] + n;
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}]; (* A294556 *)
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]   (* A296566 *)

Crossrefs

Cf. A001622, A294556, A296469, A296565.

Sequence in context: A068467 A131223 A225464 * A198213 A093766 A363368

Adjacent sequences: A296563 A296564 A296565 * A296567 A296568 A296569

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 20 2017

Status

approved