A296844 Decimal expansion of ratio-sum for A296843; see Comments.

4, 4, 6, 5, 2, 0, 5, 2, 9, 3, 4, 2, 0, 7, 5, 5, 5, 3, 6, 4, 1, 2, 6, 7, 8, 3, 7, 3, 0, 1, 3, 0, 3, 6, 0, 4, 6, 4, 1, 5, 6, 3, 1, 6, 5, 1, 8, 4, 3, 6, 4, 9, 4, 0, 1, 2, 2, 3, 2, 7, 5, 6, 0, 2, 4, 7, 3, 1, 7, 4, 6, 3, 8, 9, 3, 0, 4, 6, 8, 4, 0, 7, 6, 1, 0, 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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A296843, 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 = 4.465205293420755536412678373013036046415...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];
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}];  (* A296843 *)
GoldenRatio; s = N[Sum[-g + a[n]/a[n - 1], {n, 1, 1000}], 200];
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296844 *)

Crossrefs

Cf. A001622, A296843, A296845.

Sequence in context: A247343 A071256 A078240 * A332917 A016710 A225134

Adjacent sequences: A296841 A296842 A296843 * A296845 A296846 A296847

Keyword

nonn,easy,cons

Author

Clark Kimberling, Jan 12 2018

Status

approved