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A296460
Decimal expansion of limiting power-ratio for A296288; see Comments.
1
1, 9, 2, 0, 9, 1, 9, 8, 9, 5, 2, 2, 4, 2, 7, 6, 9, 0, 6, 6, 9, 4, 3, 2, 7, 1, 0, 0, 1, 2, 4, 4, 4, 5, 6, 5, 2, 3, 4, 9, 7, 7, 1, 2, 6, 4, 2, 0, 4, 5, 3, 3, 5, 2, 0, 3, 6, 3, 9, 3, 8, 8, 6, 4, 8, 5, 3, 6, 4, 5, 0, 7, 5, 9, 2, 9, 0, 7, 4, 9, 6, 7, 7, 7, 2, 9
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 = A296288 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 = 19.20919895224276906694327100124445652349...
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n - 1];
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}] (* A296288 *)
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] (* A296460 *)
CROSSREFS
Sequence in context: A096388 A153463 A240985 * A319533 A010160 A093962
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Dec 18 2017
STATUS
approved