A296470 Decimal expansion of limiting power-ratio for A295862; see Comments.

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

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 = A295862 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.

Formula

limiting power-ratio = 6.514710075055272773358868575343969425373...

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];
j = 1; While[j < 12, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 15}]  (* A295862 *)
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] (* A296470 *)

Crossrefs

Cf. A001622, A295862, A296469.

Sequence in context: A199724 A126743 A046613 * A195774 A320833 A216582

Adjacent sequences: A296467 A296468 A296469 * A296471 A296472 A296473

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 18 2017

Status

approved