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 1-6 of 6 results.

A278812 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278452, where c = e = 2.71828... and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

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

Views

Author

Rok Cestnik, Nov 28 2016

Keywords

Comments

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=e similar): "Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."

Examples

			1.36790126179708516966890917576048853838462452618213...

Links

Crossrefs

For sequence round(b(n)) see A278452.
For different values of c see A278808, A278809, A278810, A278811.
For b(1)=0 see A278813.

Programs

  • Mathematica
    c = E;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[ Fold[ Log[#1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

Formula

log(2*log(3*log(4*log(...)))). - Andrey Zabolotskiy, Nov 30 2016

A278450 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=4 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 14, 17, 21, 24, 27, 31, 34, 38, 41, 45, 49, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 101, 105, 109, 114, 118, 122, 127, 131, 135, 140, 144, 149, 153, 158, 162, 167, 172, 176, 181, 185, 190, 195, 200, 204, 209, 214, 218, 223, 228, 233, 238, 242, 247, 252, 257, 262, 267, 272, 277, 282, 287
Offset: 1

Views

Author

Rok Cestnik, Nov 22 2016

Keywords

Comments

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1 (due to rounding to the nearest integer a(n-1)a(n+1)/a(n)^2 is not always less than 1).
In this case b(1) = 0.4970450000... A278810. If b(1) were chosen smaller the sequence would approach 1, if it were chosen greater the sequence would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=4 similar): "Suppose one starts with b(1) = 2, the sequence would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."
b(n) = n*log_4((n+1)*log_4((n+2)*log_4(...))) ~ n*log_4(n). - Andrey Zabolotskiy, Dec 01 2016

Examples

			a(2) = round(4^0.49...) = round(1.99...) = 2.
a(3) = round(4^(1.99.../2)) = round(3.97...) = 4.
a(4) = round(4^(3.97.../3)) = round(6.28...) = 6.

Links

Crossrefs

For decimal expansion of b(1) see A278810.
For different values of c see A278448, A278449, A278451, A278452.
For b(1)=0 see A278453.

Programs

  • Mathematica
    c = 4;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
bnlist = {N[b1]};
bn = b1;
For[i = 1, i < n, ++i,
  bn = N[c^(bn/i), acc];
  If[bn > th, Break[]];
  bnlist = Append[bnlist, N[bn]];
  ];
anlist = Map[Round[#] &, bnlist]

A278808 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278448, where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

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

Views

Author

Rok Cestnik, Nov 28 2016

Keywords

Comments

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3.
No closed form expression is known. Probably transcendental but this is unproved. - Robert G. Wilson v, Dec 01 2016

Examples

			2.87188082704545465889055175504575045865652511847965...

Links

Crossrefs

For sequence round(b(n)) see A278448.
For different values of c see A278809, A278810, A278811, A278812.
For b(1)=0 see A278813.

Programs

  • Mathematica
    c = 2;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[Fold[Log2[#1*#2] &, 1, Reverse@Range[2, 144]], 10,
111][[1]] (* Robert G. Wilson v, Dec 01 2016 *)

Formula

log_2(2*log_2(3*log_2(4*log_2(...)))). - Andrey Zabolotskiy, Nov 30 2016

A278809 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278449, where c=3 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

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

Views

Author

Rok Cestnik, Nov 28 2016

Keywords

Comments

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=3 similar): "Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."

Examples

			1.08287360952073869408285031345310080257863454785385...

Links

Crossrefs

For sequence round(b(n)) see A278449.
For different values of c see A278808, A278810, A278811, A278812.
For b(1)=0 see A278813.

Programs

  • Mathematica
    c = 3;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[ Fold[ Log[3, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

Formula

log_3(2*log_3(3*log_3(4*log_3(...)))). - Andrey Zabolotskiy, Dec 01 2016

A278811 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278451, where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

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

Views

Author

Rok Cestnik, Nov 28 2016

Keywords

Comments

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=5 similar): "Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."

Examples

			0.17758191880251403338350318130866985788329770346810...

Links

Crossrefs

For sequence round(b(n)) see A278451.
For different values of c see A278808, A278809, A278810, A278812.
For b(1)=0 see A278813.

Programs

  • Mathematica
    c = 5;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[ Fold[ Log[5, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

Formula

log5(2*log5(3*log5(4*log5(...)))). - Andrey Zabolotskiy, Nov 30 2016

A278813 Decimal expansion of c in the sequence b(n+1) = c^(b(n)/n) A278453, where b(1)=0 and c is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

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

Views

Author

Rok Cestnik, Nov 28 2016

Keywords

Comments

There exists a unique value of c for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If c were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of c is found through trial and error. Suppose one starts with c = 5, the sequence b(n) would continue b(2) = 1, b(3) = 2.23..., b(4) = 3.31..., b(5) = 3.80..., b(6) = 3.39..., b(7) = 2.48..., b(8) = 1.77... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say c = 6, which gives rise to b(2) = 1, b(3) = 2.44, b(4) = 4.31..., b(5) = 6.92..., b(6) = 11.94..., b(7) = 35.38... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of c is between 5 and 6.
c satisfies 2*log_c(3*log_c(4*log_c(...))) = 1. - Andrey Zabolotskiy, Dec 02 2016
No closed form expression is known. Probably transcendental but this is unproved. - Robert G. Wilson v, Dec 02 2016

Examples

			5.75819593911037494197402886500932909247424264705531...

Links

Crossrefs

For sequence round(b(n)) see A278453.
For different values of b(1) see A278808, A278809, A278810, A278811, A278812.

Programs

  • Mathematica
    b1 = 0;
n = 100;
acc = Round[n*1.2];
th = 1000000;
c = 0;
For[p = 0, p < acc, ++p, For[d = 0, d < 9, ++d, c = c + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i, bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];];
    If[bn > th, {c = c - 1/10^p;
      Break[];
      }];
    ];
  ];
N[c, n]
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.