login
A296454
Decimal expansion of limiting power-ratio for A296257; see Comments.
1
1, 9, 9, 3, 6, 8, 4, 3, 0, 3, 3, 5, 8, 8, 2, 6, 3, 7, 3, 5, 2, 2, 3, 0, 4, 5, 3, 7, 2, 2, 4, 2, 3, 2, 9, 4, 5, 7, 5, 1, 6, 5, 4, 3, 7, 7, 7, 1, 9, 2, 7, 9, 2, 1, 0, 9, 7, 6, 4, 7, 3, 1, 3, 5, 7, 5, 6, 3, 7, 9, 9, 1, 1, 7, 1, 3, 0, 3, 3, 3, 0, 5, 5, 2, 6, 6
OFFSET
2,2
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 = A296257 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.
EXAMPLE
Limiting power-ratio = 19.93684303358826373522304537224232945751...
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]^2;
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}] (* A296257 *)
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] (* A296454 *)
CROSSREFS
Sequence in context: A140416 A340628 A226204 * A179102 A153505 A088398
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Dec 15 2017
STATUS
approved