A296469 -id:A296469 - cp's OEIS frontend

A296848 Decimal expansion of limiting power-ratio for A295953; see Comments.

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

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 = A295953, 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, A295953, A296284, A296483.

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 *)
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]   (* A296848 *)

A296471 Decimal expansion of ratio-sum for A295947; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 18 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 = A295947, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 2.427179488056039424423653103831452251757...

Cf. A001622, A295947, A296284, A296472.

a[0] = 2; a[1] = 4; b[0] = 1; b[1 ] = 3; 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}]; (* A295947 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296471 *)

A296472 Decimal expansion of limiting power-ratio for A295947; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 18 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 = A295947 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 7.171351045041137797398205678388690125273...

Cf. A001622, A296471.

a[0] = 2; a[1] = 4; b[0] = 1; b[1 ] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
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}]  (* A295947 *)
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] (* A296472 *)

A296473 Decimal expansion of ratio-sum for A295948; see Comments.

Original entry on oeis.org

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

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 = A295948, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 1.916978106743300928434323058974418608121...

Cf. A001622, A295948, A296284, A296474.

a[0] = 3; a[1] = 4; b[0] = 1; b[1 ] = 2; 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}]; (* A295948 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296473 *)

A296474 Decimal expansion of limiting power-ratio for A295948; see Comments.

Original entry on oeis.org

7, 4, 3, 2, 1, 3, 8, 6, 5, 6, 0, 2, 2, 4, 6, 4, 6, 9, 8, 6, 0, 3, 7, 4, 4, 7, 6, 2, 9, 9, 9, 1, 5, 0, 0, 0, 7, 5, 5, 1, 2, 5, 5, 0, 7, 1, 9, 6, 0, 8, 2, 8, 5, 9, 9, 8, 0, 3, 1, 0, 5, 5, 1, 5, 0, 6, 3, 4, 8, 4, 1, 8, 0, 3, 4, 0, 8, 6, 9, 6, 6, 3, 4, 8, 6, 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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296948 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 7.432138656022464698603744762999150007551...

Cf. A001622, A295948, A296473.

a[0] = 3; a[1] = 4; b[0] = 1; b[1 ] = 2; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
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}]  (* A295948 *)
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] (* A296474 *)

A296475 Decimal expansion of ratio-sum for A295949; see Comments.

Original entry on oeis.org

3, 9, 7, 5, 7, 8, 9, 5, 5, 7, 9, 7, 1, 0, 7, 3, 4, 7, 3, 4, 5, 4, 9, 6, 5, 3, 2, 6, 8, 4, 2, 9, 1, 7, 0, 7, 3, 1, 4, 4, 1, 2, 4, 7, 5, 0, 0, 4, 2, 6, 5, 2, 1, 5, 9, 1, 9, 2, 3, 7, 4, 8, 6, 6, 6, 7, 0, 5, 9, 2, 3, 2, 0, 2, 3, 6, 5, 0, 2, 4, 1, 3, 6, 7, 6, 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 = A295949, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 3.975789557971073473454965326842917073144...

Cf. A001622, A295949, A296284, A296476.

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];
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}]; (* A295949 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296475 *)

A296476 Decimal expansion of limiting power-ratio for A295949; see Comments.

Original entry on oeis.org

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

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 = A295949, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 6.136385518666220790955343949152636124460...

Cf. A001622, A295949, A296284, A296475.

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];
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}]; (* A295949 *)
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]   (* A296476 *)

Definition and sequence corrected by Clark Kimberling, Jan 05 2018

A296477 Decimal expansion of ratio-sum for A295950; see Comments.

Original entry on oeis.org

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

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 = A295950, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 4.289969273638318195483379960255534917753...

Cf. A001622, A296477, A296284, A296478.

a[0] = 1; a[1] = 4; b[0] = 2; b[1 ] = 3; 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}]; (* A295950 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296477 *)

A296478 Decimal expansion of limiting power-ratio for A295950; see Comments.

Original entry on oeis.org

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

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 = A295950, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 6.920208311693159108588213408494198705738...

Cf. A001622, A295950, A296284, A296477.

a[0] = 1; a[1] = 4; b[0] = 2; b[1 ] = 3; 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}]; (* A295950 *)
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]   (* A296478 *)

A296479 Decimal expansion of ratio-sum for A295951; see Comments.

Original entry on oeis.org

2, 5, 7, 1, 9, 7, 1, 3, 6, 9, 3, 0, 7, 5, 7, 8, 2, 4, 5, 3, 4, 0, 6, 7, 2, 4, 8, 9, 6, 7, 2, 0, 8, 7, 6, 5, 9, 8, 7, 6, 8, 8, 8, 2, 5, 4, 5, 3, 3, 3, 9, 6, 3, 8, 1, 9, 6, 8, 6, 9, 0, 7, 4, 2, 8, 4, 3, 0, 5, 5, 0, 2, 3, 7, 4, 6, 4, 3, 3, 1, 8, 4, 7, 3, 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 = A295951, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 2.571971369307578245340672489672087659876...

Cf. A001622, A295951, A296284, A296480.

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 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296479 *)

cp's OEIS Frontend

A296848 Decimal expansion of limiting power-ratio for A295953; see Comments.

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A296471 Decimal expansion of ratio-sum for A295947; see Comments.

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A296472 Decimal expansion of limiting power-ratio for A295947; see Comments.

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A296473 Decimal expansion of ratio-sum for A295948; see Comments.

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A296474 Decimal expansion of limiting power-ratio for A295948; see Comments.

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A296475 Decimal expansion of ratio-sum for A295949; see Comments.

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A296476 Decimal expansion of limiting power-ratio for A295949; see Comments.

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A296477 Decimal expansion of ratio-sum for A295950; see Comments.

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A296478 Decimal expansion of limiting power-ratio for A295950; see Comments.

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