A296469 -id:A296469 - cp's OEIS frontend

A296493 Decimal expansion of ratio-sum for A296555; see Comments.

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

Offset: 1

Clark Kimberling, Dec 19 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 = A296555, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			5.204091649313251611130187115558413050194...

Cf. A001622, A296284, A296494, A296555.

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] + 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}]; (* A296555 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296493 *)

A296491 Decimal expansion of ratio-sum for A294170; see Comments.

Original entry on oeis.org

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

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

			6.358713026984299354541477968890605504302...

Cf. A001622, A294170, A296284, A296492.

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}];  (* A294170 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296491 *)

A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.

Original entry on oeis.org

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

Offset: 2

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

			limiting power-ratio = 11.22071294787201913135639932120744822352...

Cf. A001622, A294381, A296284, A296491.

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}];  (* A294170 *)
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]   (* A296492 *)

A296496 Decimal expansion of limiting power-ratio for A294414; see Comments.

Original entry on oeis.org

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

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 = A294414, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 8.814104632202565527925188322585412678508...

Cf. A001622, A294414, A296284, A296495.

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] + b[n - 2];
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}]; (* A294414 *)
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]   (* A296496 *)

A296565 Decimal expansion of ratio-sum for A294556; see Comments.

Original entry on oeis.org

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

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 = A294556, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			4.781348441240643645893829135477090186687...

Cf. A001622, A294556, A296469, A296566.

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 - 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}]; (* A294556 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296565 *)

A296566 Decimal expansion of limiting power-ratio for A294556; see Comments.

Original entry on oeis.org

9, 0, 6, 5, 4, 5, 0, 2, 2, 6, 7, 3, 2, 3, 3, 2, 9, 2, 8, 9, 8, 9, 9, 7, 5, 9, 9, 2, 6, 2, 6, 5, 1, 5, 9, 4, 6, 1, 1, 0, 3, 9, 2, 9, 9, 7, 0, 4, 9, 3, 2, 5, 4, 2, 7, 2, 7, 0, 5, 6, 1, 1, 3, 1, 4, 2, 6, 8, 9, 2, 5, 1, 9, 5, 2, 2, 2, 6, 6, 4, 7, 1, 0, 5, 9, 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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A294556, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 9.065450226732332928989975992626515946110...

Cf. A001622, A294556, A296469, A296565.

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 - 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}]; (* A294556 *)
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]   (* A296566 *)

A296567 Decimal expansion of ratio-sum for A294557; see Comments.

Original entry on oeis.org

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

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

			2.574408888082831272283741429619184772661...

Cf. A001622, A294557, A296469, A296568.

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 *)

A296568 Decimal expansion of limiting power-ratio for A294557; see Comments.

Original entry on oeis.org

3, 6, 8, 4, 1, 4, 0, 9, 6, 3, 5, 0, 7, 1, 4, 3, 0, 4, 8, 9, 3, 4, 9, 9, 8, 7, 0, 9, 4, 3, 3, 9, 8, 3, 2, 1, 8, 5, 0, 7, 1, 3, 4, 7, 6, 0, 3, 4, 5, 1, 0, 3, 4, 9, 8, 0, 8, 8, 6, 8, 7, 6, 5, 9, 1, 0, 0, 0, 8, 3, 6, 4, 2, 0, 6, 9, 0, 2, 9, 5, 4, 8, 8, 5, 7, 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 = A294557, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 3.684140963507143048934998709433983218507...

Cf. A001622, A294557, A296469, A296567.

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 *)
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]   (* A296568 *)

A296569 Decimal expansion of ratio-sum for A294558; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 06 2018

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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 2.820339618753543689299555329148441281565...

Cf. A001622, A294558, A296469, A296570.

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 - 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}]; (* A294558 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296569 *)

A296570 Decimal expansion of limiting power-ratio for A294558; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 07 2018

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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 4.090372669767607839097909012745956501942...

Cf. A001622, A294558, A296469, A296569.

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 - 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}]; (* A294558 *)
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]   (* A296570 *)

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A296493 Decimal expansion of ratio-sum for A296555; see Comments.

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A296491 Decimal expansion of ratio-sum for A294170; see Comments.

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A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.

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A296496 Decimal expansion of limiting power-ratio for A294414; see Comments.

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A296565 Decimal expansion of ratio-sum for A294556; see Comments.

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A296566 Decimal expansion of limiting power-ratio for A294556; see Comments.

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A296567 Decimal expansion of ratio-sum for A294557; see Comments.

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A296568 Decimal expansion of limiting power-ratio for A294557; see Comments.

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A296569 Decimal expansion of ratio-sum for A294558; see Comments.

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