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 10 results.

A328913 Continued fraction expansion of A328900 = 1.50712659... solution to 2^x + 3^x = 4^x.

Original entry on oeis.org

1, 1, 1, 34, 1, 1, 2, 1, 1, 1, 2, 3, 28, 2, 1, 1, 2, 4, 3, 2, 7, 2, 35, 3, 1, 1, 2, 1, 2, 53, 1, 33, 1, 1, 1, 2, 2, 2, 35, 10, 52, 1, 1, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 1, 18, 1, 1, 7, 2, 14, 2, 84, 1, 4, 5, 3, 2, 3, 1, 2, 2, 1, 2, 40, 1, 3, 5
Offset: 0

Views

Author

M. F. Hasler, Oct 31 2019

Keywords

Comments

This number is also the solution to 1 + 1.5^x = 2^x or 1/(1 - 2^-x) = 1 + 2^-x + 3^-x, see A328900.

Examples

			A328900 = 1.50712659... = 1 + 1/(1 + 1/(1 + 1/(34 + 1/(1 + 1/(1 + 1/(2 + ...))))))

Crossrefs

Cf. A328900, A328912 (if 3 is replaced by 1).

Programs

  • Mathematica
    ContinuedFraction[ x /. FindRoot[2^x + 3^x == 4^x, {x, 1.5}, WorkingPrecision -> 100]] (* Robert G. Wilson v, Nov 12 2019 *)
  • PARI
    contfrac(solve(s=1,2,1+1.5^s-2^s)) \\ Use e.g. \p999 to get more terms.

A242208 Decimal expansion of log_2(phi), the logarithm to base 2 of phi, the "golden ratio" (1+sqrt(5))/2.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 07 2014

Keywords

Comments

The limiting fractal dimension of a pattern generated by cellular automaton rule 150 is 1+log_2(phi).
This number is also involved in the evaluation of asymptotics for the number of odd terms in Pascal's trinomial triangle.
Also, the solution to 1 + 2^x = 4^x. See A328900 for solution to 2^x + 3^x = 4^x. - M. F. Hasler, Oct 30 2019

Examples

			0.6942419136306173017387902668985952234635672852271297159809898665414...

Links

Crossrefs

Cf. A001622, A328912 (continued fraction).

Programs

  • Mathematica
    RealDigits[Log[2, GoldenRatio], 10, 100] // First
  • PARI
    print(c=log(sqrt(5)+1)/log(2)-1); digits(c\.1^default(realprecision))[^-1] \\ [^-1] to discard possibly incorrect last digit. Use e.g. \p999 to get more digits. - M. F. Hasler, Oct 30 2019

Formula

log((1 + sqrt(5))/2)/log(2).
log(sqrt(5) + 1)/log(2) - 1. - M. F. Hasler, Oct 30 2019

A328904 Decimal expansion of x = 0.7271601514124259243... solution to 1 + 3^x = 5^x.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Oct 31 2019

Keywords

Examples

			0.7271601514124259243044708440095217693545890455645833041425777641752908684323...

Crossrefs

Cf. A329334 (continued fraction).
Cf. A242208 (1 + 2^x = 4^x), A328900 (2^x + 3^x = 4^x), A328905 (1 + 2^x = 5^x).

Programs

  • Mathematica
    RealDigits[x /. FindRoot[1 + 3^x == 5^x, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 28 2023 *)
  • PARI
    print(c=solve(x=0,1, 1+3^x-5^x)); digits(c\.1^default(realprecision))[^-1] \\ [^-1] to discard possibly incorrect last digit. Use e.g. \p999 to get more digits. - M. F. Hasler, Oct 31 2019

A328905 Decimal expansion of the solution x = 0.56389552425993647949... to 1 + 2^x = 5^x.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Oct 31 2019

Keywords

Examples

			0.5638955242599364794903929459379565655152117305099552985928083801204662005228...

Crossrefs

Cf. A329334 (continued fraction).
Cf. A242208 (1 + 2^x = 4^x), A328900 (2^x + 3^x = 4^x), A328904 (1 + 3^x = 5^x).

Programs

  • Mathematica
    RealDigits[x /. FindRoot[1 + 2^x == 5^x, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 28 2023 *)
  • PARI
    print(c=solve(x=0,1, 1+2^x-5^x)); digits(c\.1^default(realprecision))[^-1] \\ [^-1] to discard possibly incorrect last digit. Use e.g. \p999 to get more digits. - M. F. Hasler, Oct 31 2019

A328907 Decimal expansion of the solution x = 0.6009668516... to 1 + 3^x = 6^x.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.6009668516136754857157052646318381206772279921330513588502639401916921204...

Crossrefs

Cf. A329337 (continued fraction).
Cf. A242208 (1 + 2^x = 4^x), A328900 (2^x + 3^x = 4^x), A328904 (1 + 3^x = 5^x), A328905 (1 + 2^x = 5^x), A328906 (1 + 2^x = 6^x).

Programs

  • Mathematica
    RealDigits[x /. FindRoot[1 + 3^x == 6^x, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 28 2023 *)
  • PARI
    print(c=solve(x=0,1, 1+3^x-6^x)); digits(c\.1^default(realprecision))[^-1] \\ [^-1] to discard possibly incorrect last digit. Use e.g. \p999 to get more digits. - M. F. Hasler, Oct 31 2019

A328906 Decimal expansion of the solution x = 0.4895363211996... to 1 + 2^x = 6^x.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.489536321199649488689875316822650189403586515771912127846436678619255628...

Crossrefs

Cf. A329336 (continued fraction).
Cf. A242208 (1 + 2^x = 4^x), A328900 (2^x + 3^x = 4^x), A328904 (1 + 3^x = 5^x), A328905 (1 + 2^x = 5^x), A328907 (1 + 3^x = 6^x).

Programs

  • Mathematica
    RealDigits[x /. FindRoot[1 + 2^x == 6^x, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 28 2023 *)
  • PARI
    print(c=solve(x=0,1, 1+2^x-6^x)); digits(c\.1^default(realprecision))[^-1] \\ [^-1] to discard possibly incorrect last digit. Use e.g. \p999 to get more digits. - M. F. Hasler, Oct 31 2019

A329334 Continued fraction of A328904 = 0.7271601514124259..., solution to 1 + 3^x = 5^x.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 72, 1, 3, 2, 6, 1, 1, 2, 45, 1, 5, 13, 73, 1, 2, 1, 9, 1, 1, 1, 3, 2, 3, 3, 2, 1, 2, 2, 1, 1, 19, 1, 1, 1, 1, 5, 1, 5, 2, 4, 3, 1, 6, 1, 1, 2, 1, 9, 8, 1, 4, 1, 1, 20, 1, 1, 2, 1, 5, 2, 2, 1, 2, 5, 1, 56, 1, 1, 1, 6, 127, 1, 1, 7, 2, 7, 1, 6, 1, 1, 3, 1, 54, 1, 1, 3, 2, 1, 1, 3
Offset: 0

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.7271601514124259... = 0 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(1  + 1/(72 + 1/...))))))

Crossrefs

Cf. A328912 (cont. frac. of A242208: 1 + 2^x = 4^x), A328913 (cont. frac. of A328900: 2^x + 3^x = 4^x), A329337 (cont. frac. of A328907: 1 + 3^x = 6^x).

Programs

  • PARI
    contfrac(c=solve(x=0,1, 1+3^x-5^x))[^-1] \\ discarding possibly incorrect last term. Use e.g. \p999 to get more terms. - M. F. Hasler, Oct 31 2019

A329337 Continued fraction of A328907 = 0.6009668516..., solution to 1 + 3^x = 6^x.

Original entry on oeis.org

0, 1, 1, 1, 1, 40, 1, 3, 2, 1, 2, 23, 1, 13, 1, 8, 1, 15, 2, 3, 103, 4, 10, 4, 2, 2, 2, 1, 1, 84, 1, 4, 1, 3, 1, 1, 5, 1, 7, 23, 8, 1, 8, 24, 1, 1, 2, 12, 39, 14, 19, 3, 4, 8, 3, 2, 1, 4, 1, 8, 1, 1, 1, 2, 1, 10, 1, 35, 1, 10, 2, 2, 2, 1, 1, 15, 2, 3, 1, 4, 7, 5, 1, 9, 1, 1, 1, 1, 2, 3, 3, 2, 1, 4, 54, 4, 1, 3, 2, 1, 1, 1, 1, 4, 22, 1, 4, 3, 1, 1, 1, 2, 6, 1, 1, 4, 1, 8, 1, 20
Offset: 0

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.6009668516... = 0 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(40 + 1/(1 + 1/(3 + ...)))))))

Crossrefs

Cf. A328912 (cont. frac. of A242208: 1 + 2^x = 4^x), A328913 (cont. frac. of A328900: 2^x + 3^x = 4^x), A329335 (cont. frac. of A328905: 1 + 2^x = 5^x).

Programs

  • Mathematica
    ContinuedFraction[x/.FindRoot[1+3^x==6^x,{x,1},WorkingPrecision->150]] (* Harvey P. Dale, Jun 13 2022 *)
  • PARI
    contfrac(c=solve(x=0,1, 1+3^x-6^x))[^-1] \\ discarding possibly incorrect last term. Use e.g. \p999 to get more terms. - M. F. Hasler, Oct 31 2019

A329336 Continued fraction of A328906 = 0.4895363211996..., solution to 1 + 2^x = 6^x.

Original entry on oeis.org

0, 2, 23, 2, 1, 1, 4, 1, 1, 27, 4, 12, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 2, 6, 1, 10, 4, 3, 4, 1, 2, 1, 1, 43, 69, 1, 2, 41, 1, 3, 2, 3, 3, 1, 5, 4, 1, 1, 1, 7, 1, 1, 1, 11, 13, 2, 3, 1, 1, 1, 118, 2, 1, 1, 12, 1, 2, 2, 2, 6, 2, 3, 1, 4, 1, 8, 1, 1, 18, 2, 21, 1, 4, 1, 3, 1, 51, 6, 1, 1, 18, 2, 1, 1, 2, 56, 1, 1, 5, 4, 1, 4, 7, 1, 2, 2, 1, 9, 76, 2, 1, 3, 1, 5, 3, 1, 7, 6
Offset: 0

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.4895363211996... = 0 + 1/(2 + 1/(23 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + 1/...))))))

Crossrefs

Cf. A328912 (cont. frac. of A242208: 1 + 2^x = 4^x), A328913 (cont. frac. of A328900: 2^x + 3^x = 4^x), A329334 (cont. frac. of A328904: 1 + 3^x = 5^x).

Programs

  • Mathematica
    ContinuedFraction[x/.FindRoot[1+2^x==6^x,{x,.4},WorkingPrecision->1000],150] (* Harvey P. Dale, Oct 15 2022 *)
  • PARI
    contfrac(c=solve(x=0,1, 1+2^x-6^x))[^-1] \\ discarding possibly incorrect last term. Use e.g. \p999 to get more terms. - M. F. Hasler, Oct 31 2019

A329335 Continued fraction of A328905 = 0.5638955242599..., solution to 1 + 2^x = 5^x.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 2, 1, 3, 3, 1, 5, 3, 1, 1, 3, 1, 1, 6, 1, 36, 20, 18, 3, 1, 3, 2, 1, 2, 9, 3, 2, 1, 1, 1, 2, 7, 1, 1, 5, 1, 112, 2, 1, 6, 2, 1, 1, 1, 1, 2, 44, 1, 2, 3, 70, 1, 1, 1, 12, 3, 1, 5, 6, 1, 1, 10, 4, 4, 2, 3, 1, 7, 1, 4, 1, 1, 1, 5, 2, 1, 5, 1, 4, 3, 1, 1, 1, 1, 2, 1, 1, 4, 6, 7, 2
Offset: 0

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.5638955242599... = 0 + 1/(1 + 1/(1 + 1/(3 + 1/(2 + 1/(2  + 1/(2 + 1/...))))))

Crossrefs

Cf. A328912 (cont. frac. of A242208: 1 + 2^x = 4^x), A328913 (cont. frac. of A328900: 2^x + 3^x = 4^x), A329337 (cont. frac. of A328907: 1 + 3^x = 6^x).

Programs

  • PARI
    contfrac(c=solve(x=0,1, 1+2^x-5^x))[^-1] \\ discarding possibly incorrect last term. Use e.g. \p999 to get more terms. - M. F. Hasler, Oct 31 2019
Showing 1-10 of 10 results.

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