A296452 Decimal expansion of limiting power-ratio for A296245; see Comments.

3, 5, 9, 2, 9, 5, 4, 9, 2, 5, 5, 5, 8, 3, 1, 8, 4, 0, 9, 0, 2, 1, 6, 6, 6, 8, 7, 8, 3, 5, 1, 2, 1, 9, 1, 3, 2, 0, 7, 1, 5, 1, 8, 3, 9, 7, 5, 7, 9, 0, 8, 5, 6, 0, 7, 0, 8, 3, 0, 3, 1, 7, 9, 1, 0, 5, 2, 3, 9, 2, 8, 0, 5, 5, 2, 9, 5, 3, 9, 2, 1, 7, 7, 5, 4, 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 = A296245 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.

Links

Table of n, a(n) for n=2..87.

Example

Limiting power-ratio = 35.92954925558318409021666878351219132071...

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] + b[n]^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}]  (* A296245 *)
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] (* A296452 *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A186190 A019739 A101298 * A225594 A210946 A319984

Adjacent sequences: A296449 A296450 A296451 * A296453 A296454 A296455

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 15 2017

Status

approved