A296434 -id:A296434 - cp's OEIS frontend

A296428 Decimal expansion of ratio-sum for A295367; see Comments.

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

Offset: 1

Clark Kimberling, Dec 14 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 = A295367, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			8.83944262127645153974944...

Cf. A001622, A295367.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[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}]; (* A295367 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296428 *)

A296429 Decimal expansion of ratio-sum for A296266; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 14 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 = A296266, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			10.24289391012130367983487...

Cf. A001622, A296266.

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]*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}]; (* A296266 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296429 *)

A296430 Decimal expansion of ratio-sum for A296272; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 14 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 = A296272 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			ratio-sum = 12.5831861005560957189096...

Cf. A001622, A296272.

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];
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}]; (* A296272 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296430 *)

A296432 Decimal expansion of ratio-sum for A296284; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 15 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 = A296284 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Ratio-sum = 6.21032710946618494227967...

Cf. A001622, A296284.

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]; (* A296284 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296432 *)

A296433 Decimal expansion of ratio-sum for A296288; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 15 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 = A296288 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Ratio-sum = 7.093883244558233282518632...

Cf. A001622, A296288.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]; (* A296288 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296433 *)

A296453 Decimal expansion of limiting power-ratio for A296251; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 15 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 = A296251 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Limiting power-ratio = 27.09717363716137318861436923160085309363...

Cf. A001622, A296251.

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]^2;
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 15}]  (* A296251 *)
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] (* A296453 *)

A296454 Decimal expansion of limiting power-ratio for A296257; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 15 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 = A296257 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Limiting power-ratio = 19.93684303358826373522304537224232945751...

Cf. A001622, A296257.

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]^2;
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 15}]  (* A296257 *)
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] (* A296454 *)

A296455 Decimal expansion of limiting power-ratio for A296260; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 15 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 = A296260 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Limiting power-ratio = 23.10815724358788604144450707514353840694...

Cf. A001622, A296260.

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 < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 15}]  (* A296260 *)
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] (* A296455 *)

A296456 Decimal expansion of limiting power-ratio for A296266; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 15 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 = A296266 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Limiting power-ratio = 26.38762899138511775383320781232073665030...

Cf. A001622, A296266.

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]*b[n - 2];
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 15}]  (* A296266 *)
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] (* A296456 *)

A296457 Decimal expansion of limiting power-ratio for A296272; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 15 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 = A296272 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Limiting power-ratio = 31.14503268621267806442242062631447330734...

Cf. A001622, A296272.

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]*b[n - 1];
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 15}]  (* A296272 *)
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] (* A296457 *)

cp's OEIS Frontend

A296428 Decimal expansion of ratio-sum for A295367; see Comments.

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A296429 Decimal expansion of ratio-sum for A296266; see Comments.

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A296430 Decimal expansion of ratio-sum for A296272; see Comments.

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A296432 Decimal expansion of ratio-sum for A296284; see Comments.

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A296433 Decimal expansion of ratio-sum for A296288; see Comments.

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A296453 Decimal expansion of limiting power-ratio for A296251; see Comments.

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A296454 Decimal expansion of limiting power-ratio for A296257; see Comments.

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A296455 Decimal expansion of limiting power-ratio for A296260; see Comments.

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A296456 Decimal expansion of limiting power-ratio for A296266; see Comments.

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