A296504 Decimal expansion of limiting power-ratio for A294553; see Comments.

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

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 = A294553, 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 = 6.136385518707822221768412790440089960037...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
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}]; (* A294553 *)
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]   (* A296504 *)

Crossrefs

Cf. A001622, A294553, A296469, A296503.

Sequence in context: A321093 A219552 A296476 * A360779 A188859 A074584

Adjacent sequences: A296501 A296502 A296503 * A296505 A296506 A296507

Keyword

nonn,easy,cons

Author

Clark Kimberling, Apr 14 2018

Status

approved