A298172 Decimal expansion of limiting power-ratio for A296776; see Comments.

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

Offset

2,3

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 = A296776, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.

Links

Table of n, a(n) for n=2..87.

Example

limiting power-ratio = 11.65837748505646668170868260629394947392...

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 n;
j = 1; While[j < 16, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A296776 *)
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]   (* A298172 *)

Crossrefs

Cf. A001622, A296462, A296776.

Sequence in context: A147313 A242759 A021607 * A225113 A329247 A133618

Adjacent sequences: A298169 A298170 A298171 * A298173 A298174 A298175

Keyword

nonn,easy,cons

Author

Clark Kimberling, Feb 09 2018

Status

approved