A296284 -id:A296284 - cp's OEIS frontend

A296499 Decimal expansion of ratio-sum for A294546; see Comments.

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

Offset: 1

Clark Kimberling, Dec 20 2017

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 = A294546, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			4.737057998792217365428375621295759326159...

Cf. A001622, A294546, A296284, A296500.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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}]; (* A294549 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296499 *)

A296500 Decimal expansion of limiting power-ratio for A294546; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 20 2017

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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A294546, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 7.920208071319294432128690529988578214476...

Cf. A001622, A294546, A296284, A296499.

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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}]; (* A294546 *)
z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120]   (* A296500 *)

cp's OEIS Frontend

A296499 Decimal expansion of ratio-sum for A294546; see Comments.

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