A296284 -id:A296284 - cp's OEIS frontend

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

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 *)

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 *)

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

Original entry on oeis.org

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 *)

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A296473 Decimal expansion of ratio-sum 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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A296479 Decimal expansion of ratio-sum for A295951; see Comments.

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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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