A298171 Decimal expansion of ratio-sum for A296776; see Comments.

5, 6, 1, 1, 5, 6, 2, 0, 5, 7, 6, 5, 6, 2, 2, 5, 4, 7, 7, 0, 3, 2, 4, 4, 3, 4, 5, 6, 0, 9, 2, 5, 7, 9, 4, 8, 0, 9, 8, 2, 7, 0, 9, 5, 8, 6, 5, 5, 5, 5, 7, 3, 7, 0, 6, 5, 0, 1, 9, 0, 5, 7, 3, 9, 5, 3, 5, 1, 0, 5, 4, 3, 3, 1, 7, 6, 6, 7, 6, 0, 2, 0, 1, 0, 5, 4

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 = A298171, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.

Links

Table of n, a(n) for n=1..86.

Example

ratio-sum = 5.611562057656225477032443456092579480982...

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] + 2 n;
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}];  (* A296776 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A298171 *)

Crossrefs

Cf. A001622, A296462, A296776.

Sequence in context: A243108 A355186 A287610 * A046614 A371048 A080130

Adjacent sequences: A298168 A298169 A298170 * A298172 A298173 A298174

Keyword

nonn,easy,cons

Author

Clark Kimberling, Feb 09 2018

Status

approved