A296469 -id:A296469 - cp's OEIS frontend

A296480 Decimal expansion of limiting power-ratio for A295951; see Comments.

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

Offset: 1

Clark Kimberling, Jan 05 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 = A295951, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 6.749918662049985424828699465394565293987...

Cf. A001622, A295951, A296284, A296479.

a[0] = 2; a[1] = 3; b[0] = 1; b[1 ] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[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}]; (* A295951 *)
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]   (* A296480 *)

A296481 Decimal expansion of ratio-sum for A295952; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 05 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 = A295952, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 4.845853683514362071350020567372501789034...

Cf. A001622, A296284, A296482.

a[0] = 1; a[1] = 5; b[0] = 2; b[1 ] = 3; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[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}]; (* A295952 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296481 *)

A296482 Decimal expansion of limiting power-ratio for A295952; see Comments.

Original entry on oeis.org

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

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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A295952, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 7.090700687355142881167747526503371215921...

Cf. A001622, A295952, A296284, A296481.

a[0] = 1; a[1] = 5; b[0] = 2; b[1 ] = 3; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[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}]; (* A295952 *)
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]   (* A296482 *)

A296483 Decimal expansion of ratio-sum for A295953; see Comments.

Original entry on oeis.org

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

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 = A295953, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 4.194678679056371758991840818123954420964...

Cf. A001622, A295953, A296284, A296848.

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;
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}]; (* A295953 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296483 *)

A296485 Decimal expansion of ratio-sum for A295960; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 12 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 = A295960, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 3.556958436660903995097548494451097938728...

Cf. A001622, A295960, A296284, A296486.

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;
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}]; (* A295960 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296485 *)

A296486 Decimal expansion of limiting power-ratio for A295960; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 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 = A295960, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 5.859996629844644031286035775860542608816...

Cf. A001622, A295960, A296284, A296485.

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;
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}]; (* A295960 *)
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]   (* A296486 *)

A296487 Decimal expansion of ratio-sum for A293076; see Comments.

Original entry on oeis.org

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

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 = A293076, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 3.092262285773106330183534655202716162425...

Cf. A001622, A293076, A296284, A296488.

a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 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}]; (* A293076 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296487 *)

A296488 Decimal expansion of limiting power-ratio for A293076; see Comments.

Original entry on oeis.org

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

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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A293076, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 4.863698868156079195859888752149657198717...

Cf. A001622, A293076, A296284, A296487.

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];
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}]; (* A293076 *)
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]   (* A296488 *)

A296489 Decimal expansion of ratio-sum for A293358; see Comments.

Original entry on oeis.org

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

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 = A293358, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 3.535917393515259350716802232248550673355...

Cf. A001622, A293358, A296284, A296490.

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];
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}]; (* A293358 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296489 *)

A296490 Decimal expansion of limiting power-ratio for A293358; see Comments.

Original entry on oeis.org

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

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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A293358, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 5.792497484521075892747738075882867016822...

Cf. A001622, A293358, A296284, A296489.

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];
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}]; (* A293358 *)
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]   (* A296490 *)

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A296480 Decimal expansion of limiting power-ratio for A295951; see Comments.

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A296481 Decimal expansion of ratio-sum for A295952; see Comments.

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A296482 Decimal expansion of limiting power-ratio for A295952; see Comments.

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A296483 Decimal expansion of ratio-sum for A295953; see Comments.

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A296485 Decimal expansion of ratio-sum for A295960; see Comments.

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A296486 Decimal expansion of limiting power-ratio for A295960; see Comments.

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A296487 Decimal expansion of ratio-sum for A293076; see Comments.

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A296488 Decimal expansion of limiting power-ratio for A293076; see Comments.

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A296489 Decimal expansion of ratio-sum for A293358; see Comments.

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