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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 = A294170, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

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Decimal expansion of ratio-sum for A296776A294170; see 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 = A296776, A294170, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

u = Table[a[n], {n, 0, k}]; (* A296776 A294170 *)

Cf. A001622, A294381, A294170, A296284, A296492.

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allocated Decimal expansion of ratio-sum for Clark KimberlingA296776; see Comments.

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

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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 = A296776, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

6.358713026984299354541477968890605504302...

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 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] (* A296491 *)

Cf. A001622, A294381, A296284, A296492.

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nonn,easy,cons

Clark Kimberling, Dec 20 2017

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