A296284 -id:A296284 - cp's OEIS frontend

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

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 *)

A296494 Decimal expansion of limiting power-ratio for A296555; see Comments.

Original entry on oeis.org

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

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

			limiting power-ratio = 8.643486886551748831682988666251436446979...

Cf. A001622, A296284, A296493, 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 *)
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]   (* A296494 *)

A296495 Decimal expansion of ratio-sum for A294414; see Comments.

Original entry on oeis.org

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

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.

			4.585197174184565401988869057817650746901...

Cf. A001622, A294414, A296284, A296496.

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 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296495 *)

A296497 Decimal expansion of ratio-sum for A294541; see Comments.

Original entry on oeis.org

3, 4, 8, 9, 8, 0, 1, 8, 3, 9, 1, 6, 5, 1, 8, 0, 8, 8, 3, 9, 3, 9, 5, 8, 4, 0, 8, 2, 8, 8, 2, 5, 7, 1, 7, 2, 3, 7, 6, 9, 1, 6, 5, 8, 1, 9, 0, 7, 5, 4, 7, 8, 7, 6, 4, 0, 7, 4, 9, 9, 5, 3, 8, 4, 0, 5, 7, 0, 2, 8, 6, 8, 7, 8, 0, 0, 5, 7, 0, 3, 3, 7, 8, 9, 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 = A294541, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			3.489801839165180883939584082882571723769...

Cf. A001622, A294541, A296284, A296498.

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] + 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}]; (* A294541 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296497 *)

A296498 Decimal expansion of limiting power-ratio for A294541; see Comments.

Original entry on oeis.org

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

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 = A294541, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 5.416437430138486313848494950880888008486...

Cf. A001622, A294541, A296284, A296497.

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}]; (* A294541 *)
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]   (* A296498 *)

cp's OEIS Frontend

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

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A296494 Decimal expansion of limiting power-ratio for A296555; see Comments.

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A296495 Decimal expansion of ratio-sum for A294414; see Comments.

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A296497 Decimal expansion of ratio-sum for A294541; see Comments.

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