A296847 Decimal expansion of ratio-sum for A296846; see Comments.

2, 5, 1, 3, 4, 8, 8, 7, 0, 3, 3, 6, 6, 4, 2, 1, 5, 6, 2, 0, 4, 4, 5, 4, 9, 0, 9, 4, 9, 3, 9, 1, 3, 9, 1, 5, 2, 1, 9, 1, 7, 5, 6, 9, 4, 4, 3, 0, 5, 3, 6, 7, 3, 0, 6, 5, 3, 1, 7, 8, 9, 8, 7, 7, 2, 3, 6, 5, 3, 9, 9, 9, 5, 2, 4, 6, 1, 8, 4, 0, 4, 0, 7, 2, 9, 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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A296846, 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=1..86.

Example

ratio-sum = 2.513488703366421562044549094939139152191...

Mathematica

a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] - b[n - 2];
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A296846 *)
g = GoldenRatio; s = N[Sum[g - a[n]/a[n - 1], {n, 1, 1000}], 200];  (* A296847 *)
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296847 *)

Crossrefs

Cf. A001622, A296846, A296848.

Sequence in context: A011456 A233384 A186692 * A100084 A269598 A100226

Adjacent sequences: A296844 A296845 A296846 * A296848 A296849 A296850

Keyword

nonn,easy,cons

Author

Clark Kimberling, Jan 12 2018

Status

approved