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A296498
Decimal expansion of limiting power-ratio for A294541; see Comments.
3
5, 4, 1, 6, 4, 3, 7, 4, 3, 0, 1, 3, 8, 4, 8, 6, 3, 1, 3, 8, 4, 8, 4, 9, 4, 9, 5, 0, 8, 8, 0, 8, 8, 8, 0, 0, 8, 4, 8, 6, 7, 6, 6, 5, 9, 9, 0, 1, 5, 3, 5, 0, 6, 5, 3, 2, 7, 4, 6, 4, 2, 8, 1, 0, 9, 3, 6, 4, 1, 1, 5, 9, 4, 1, 6, 6, 6, 0, 3, 7, 2, 9, 1, 4, 5, 2
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 = A294541, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
EXAMPLE
limiting power-ratio = 5.416437430138486313848494950880888008486...
MATHEMATICA
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
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}]; (* A294541 *)
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] (* A296498 *)
CROSSREFS
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Dec 20 2017
STATUS
approved