cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 24 results. Next

A296469 Decimal expansion of ratio-sum for A295862; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 18 2017

Keywords

Comments

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 = A295862, 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. Guide to more ratio-sums and limiting power-ratios:
****
Sequence A ratio-sum for A limiting power-ratio for A
A295862 A296469 A296470
A295947 A296471 A296472
A295948 A296473 A296474
A295949 A296475 A296476
A295950 A296477 A296478
A295951 A296479 A296480
A295952 A296481 A296482
A295953 A296483 A296848
A295960 A296485 A296486
A293076 A296487 A296488
A293358 A296489 A296490
A294170 A296491 A296492
A296555 A296493 A296494
A294414 A296495 A296496
A294541 A296497 A296498
A294546 A296499 A296500
A294552 A296501 A296494
A296776 A298171 A298172
A294553 A296503 A296504
A296556 A296565 A296566
A296557 A296567 A296568
A296558 A296569 A296570

Examples

			ratio-sum = 6.21032710946618494227967...

Crossrefs

Cf. A001622, A296284, A296470.

Programs

  • Mathematica
    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];
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}]; (* A295862 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296469 *)

A296425 Decimal expansion of ratio-sum for A296245; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 14 2017

Keywords

Comments

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 = A296245, 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.

Examples

			14.9763271448563041241168963...

Crossrefs

Cf. A001622, A296245.

Programs

  • Mathematica
    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;
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}]; (* A296245 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296425 *)

A296434 Decimal expansion of ratio-sum for A296292; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2017

Keywords

Comments

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 = A296292 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.

Examples

			Ratio-sum = 8.012968903095661472515541...

Crossrefs

Cf. A001622, A296292.

Programs

  • Mathematica
    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] + n*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}]; (* A296292 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296434 *)

A296461 Decimal expansion of limiting power-ratio for A296292; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 18 2017

Keywords

Comments

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 = A296292 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.

Examples

			limiting power-ratio = 21.74130735523558735581498585908915856896...

Crossrefs

Cf. A001622, A296292.

Programs

  • Mathematica
    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] + n*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}]  (* A296292 *)
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] (* A296461 *)

A296844 Decimal expansion of ratio-sum for A296843; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 12 2018

Keywords

Comments

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 = A296843, 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.

Examples

			ratio-sum = 4.465205293420755536412678373013036046415...

Crossrefs

Cf. A001622, A296843, A296845.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A296843 *)
GoldenRatio; s = N[Sum[-g + a[n]/a[n - 1], {n, 1, 1000}], 200];
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296844 *)

A296850 Decimal expansion of ratio-sum for A296849; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 12 2018

Keywords

Comments

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 = A296849, we have g = 1 + sqrt(2). See A296425..A296434 for related ratio-sums and A296452..A296461 for related limiting power-ratios.

Examples

			ratio-sum = 2.898430373605188679364554049974531941517...

Crossrefs

Cf. A296849.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 8, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296849 *)
r = 1 + Sqrt[2]; s = N[Sum[-r + a[n]/a[n - 1], {n, 1, 1000}], 200];
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296850 *)

A296847 Decimal expansion of ratio-sum for A296846; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 12 2018

Keywords

Comments

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 = A296846, 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.

Examples

			ratio-sum = 2.513488703366421562044549094939139152191...

Crossrefs

Cf. A001622, A296846, A296848.

Programs

  • Mathematica
    a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] - b[n - 2];
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A296846 *)
g = GoldenRatio; s = N[Sum[g - a[n]/a[n - 1], {n, 1, 1000}], 200];  (* A296847 *)
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296847 *)

A297012 Decimal expansion of ratio-sum for A297011; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 13 2018

Keywords

Comments

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 = A297011, we have g = 1 + sqrt(2). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

Examples

			ratio-sum = 2.518439346012623900896837641191550691016...

Crossrefs

Cf. A297011.

Programs

  • Mathematica
    a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n];
j = 1; While[j < 9, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A297011 *)
r = 1 + Sqrt[2]; s = N[Sum[r - a[n]/a[n - 1], {n, 1, 1000}], 200];
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A297012 *)

A296426 Decimal expansion of ratio-sum for A296251; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 14 2017

Keywords

Comments

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 = 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.

Examples

			10.6856696299018933209586091...

Crossrefs

Cf. A001622, A296251.

Programs

  • Mathematica
    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 < 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}]; (* A296246 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296426 *)

A296427 Decimal expansion of ratio-sum for A296257; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 14 2017

Keywords

Comments

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 = 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.

Examples

			7.48659823886117515083042...

Crossrefs

Cf. A001622, A296257.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]^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}]; (* A296257 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296427 *)
Showing 1-10 of 24 results. Next

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