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A296570
Decimal expansion of limiting power-ratio for A294558; see Comments.
4
4, 0, 9, 0, 3, 7, 2, 6, 6, 9, 7, 6, 7, 6, 0, 7, 8, 3, 9, 0, 9, 7, 9, 0, 9, 0, 1, 2, 7, 4, 5, 9, 5, 6, 5, 0, 1, 9, 4, 2, 5, 4, 5, 6, 6, 2, 3, 2, 1, 8, 6, 8, 0, 7, 8, 1, 2, 1, 0, 8, 9, 7, 5, 6, 5, 4, 9, 6, 7, 0, 3, 7, 7, 2, 7, 0, 0, 1, 0, 2, 0, 2, 7, 8, 2, 0
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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
EXAMPLE
limiting power-ratio = 4.090372669767607839097909012745956501942...
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 - 1] + b[n - 2] - n;
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}]; (* A294558 *)
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] (* A296570 *)
CROSSREFS
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Jan 07 2018
STATUS
approved