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.

Previous Showing 11-20 of 20 results.

A298514 Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n + 1)*g), where g = (1 + sqrt (5))/2, s(n) = (s(n - 1) + 1)^(1/2), s(0) = 3.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = golden ratio, A001622. See A298512 for a guide to related sequences.

Examples

			s(n) -> g = (1+sqrt(5))/2, as at A001622.
s(0) + s(1) + ... + s(n) - (n + 1)*g -> 1.928338346029993604661257220820526670...

Crossrefs

Cf. A001622, A298512, A298513.

Programs

  • Mathematica
    s[0] = 3; d = 1; p = 1/2; s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
z = 200 ; g = GoldenRatio; s = N[-(z + 1)*g + Sum[s[n], {n, 0, z}], 150 ];
RealDigits[s, 10][[1]];  (* A298514 *)

A298515 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) -s(1) - ... - s(n)), where g = (1 + sqrt (3))/2, s(n) = (s(n - 1) + 1/2)^(1/2), s(0) = 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 11 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = (1 + sqrt (3))/2. See A298512 for a guide to related sequences.

Examples

			s(n) -> g = (1+sqrt(3))/2.
(n+1)*g - s(0) - s(1) - ... - s(n) -> 0.590731681518077574185979517684194787...

Crossrefs

Cf. A298512, A298516, A298517.

Programs

  • Mathematica
    s[0] = 1; d = 1/2; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]];
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
StringJoin[StringTake[ToString[s], 41], "..."]
RealDigits[s, 10][[1]]   (* A298515 *)

A298524 Decimal expansion of lim_ {n->oo} ((n+1)*g - s(0) - s(1) - ... - s(n)), where g = 1.790044015672757..., s(n) = (s(n - 1) + sqrt(2))^(1/2), s(0) = 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = positive zero of x^2 - x - sqrt(2). See A298512 for a guide to related sequences.

Examples

			(n+1)*g - s(0) - s(1) - ... - s(n) -> 1.119791807304608296546488960103955059141...

Crossrefs

Cf. A298512, A298525, A298526.

Programs

  • Mathematica
    s[0] = 1; d = Sqrt[2]; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298524 *)

A298525 Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n+1)*g), where g = 1.790044015672757..., s(n) = (s(n - 1) + sqrt(2))^(1/2), s(0) = 2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = positive zero of x^2 - x - sqrt(2). See A298512 for a guide to related sequences.

Examples

			s(0) + s(1) + ... + s(n) - (n+1)*g -> 0.289932185664479574074147512346151058813...

Crossrefs

Cf. A298512, A298524.

Programs

  • Mathematica
    s[0] = 2; d = Sqrt[2]; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[- g + s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298525 *)

A298527 Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n+1)*g), where g = 1.9078532620869538..., s(n) = (s(n - 1) + sqrt(3))^(1/2), s(0) = 2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = positive zero of x^2 - x - sqrt(3). See A298512 for a guide to related sequences.

Examples

			s(0) + s(1) + ... + s(n) - (n+1)*g -> 0.124653138346990958325743855236362833575...

Crossrefs

Cf. A298512, A298526.

Programs

  • Mathematica
    s[0] = 2; d = Sqrt[3]; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[- g + s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298527 *)

A298528 Decimal expansion of lim_{n->oo} ((n+1)*g - s(0) - s(1) - ... - s(n)), where g = 2.22287022972104..., s(n) = (s(n - 1) + e)^(1/2), s(0) = 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2018

Keywords

Comments

Limit_{n->oo} s(n) = g = positive zero of x^2 - x - e. See A298512 for a guide to related sequences.

Examples

			1.604332641724577300539659547213826891...

Crossrefs

Cf. A298512, A298529.

Programs

  • Mathematica
    s[0] = 1; d = E; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298528 *)

A298529 Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n+1)*g), where g = 2.22287022972104..., s(n) = (s(n - 1) + e)^(1/2), s(0) = 2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = positive zero of x^2 - x - e. See A298512 for a guide to related sequences.

Examples

			s(0) + s(1) + ... + s(n) - (n+1)*g -> 0.6355719693699384520319618420986777...

Crossrefs

Cf. A298512, A298528.

Programs

  • Mathematica
    s[0] = 2; d = E; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298529 *)

A298530 Decimal expansion of lim_ {n->oo} ((n+1)*g - s(0) - s(1) - ... - s(n)), where g = 2.3416277185114784317..., s(n) = (s(n - 1) + Pi)^(1/2), s(0) = 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = positive zero of x^2 - x - Pi. See A298512 for a guide to related sequences.

Examples

			lim_ {n->oo} ((n+1)*g-s(0)-s(1)-...-s(n)) -> 1.732644070061485804124716667...

Crossrefs

Cf. A298512, A298531.

Programs

  • Mathematica
    s[0] = 1; d = Pi; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298530 *)

A298521 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = 2.1478990357..., s(n) = (s(n - 1) + 1)^(2/3), s(0) = 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = real zero of x^(2/3) - x - 1. See A298512 for a guide to related sequences.

Examples

			((n + 1)*g - s(0) - s(1) - ... - s(n)) -> 2.19524431421173816419739210020867...

Crossrefs

Cf. A298512, A298520.

Programs

  • Mathematica
    s[0] = 1; d = 1; p = 2/3;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298521 *)

A298532 Decimal expansion of lim_ {n->oo} ((n+1)*g - s(0) - s(1) - ... - s(n)), where g = 1.866760399173862092990..., s(n) = (s(n - 1) + tau)^(1/2), s(0) = tau = (1+sqrt(5))/2 (golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 13 2018

Keywords

Comments

(lim_ {n->oo} s(n)) = g = positive zero of x^2 - x - tau. See A298512 for a guide to related sequences.

Examples

			lim_ {n->oo} ((n+1)*g - s(0) - s(1) - ... - s(n)) -> 0.3415333980529057321950...

Crossrefs

Cf. A298512, A298531.

Programs

  • Mathematica
    tau = GoldenRatio;
s[0] = tau; d = tau; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298532 *)
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