A296567 - OEIS

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A296567

Decimal expansion of ratio-sum for A294557; see Comments.

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

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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 = A294557, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

LINKS

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

EXAMPLE

2.574408888082831272283741429619184772661...

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 - 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}]; (* A294557 *)

g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]

Take[RealDigits[s, 10][[1]], 100] (* A296567 *)