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A296461
Decimal expansion of limiting power-ratio for A296292; see Comments.
25
2, 1, 7, 4, 1, 3, 0, 7, 3, 5, 5, 2, 3, 5, 5, 8, 7, 3, 5, 5, 8, 1, 4, 9, 8, 5, 8, 5, 9, 0, 8, 9, 1, 5, 8, 5, 6, 8, 9, 6, 3, 3, 2, 1, 7, 2, 8, 0, 7, 1, 9, 6, 3, 7, 5, 6, 3, 3, 6, 9, 0, 1, 3, 3, 8, 3, 5, 5, 4, 4, 6, 2, 2, 8, 6, 5, 5, 8, 3, 9, 8, 9, 6, 2, 9, 6
OFFSET
2,1
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 = A296292 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 = 21.74130735523558735581498585908915856896...
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}] (* A296292 *)
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] (* A296461 *)
CROSSREFS
Sequence in context: A194797 A255138 A115629 * A144696 A072248 A317360
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Dec 18 2017
STATUS
approved