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 1-2 of 2 results.

A091667 Decimal expansion of ((-1-sqrt(5))/2 + sqrt((5+sqrt(5))/2))*e^((2*Pi)/5).

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jan 27 2004

Keywords

Comments

Has a nice (non-simple) continued fraction due to Ramanujan.
Continued fraction is 1/(1+q/(1+q^2/(1+q^3/(1+...)))) where q=exp(-2*Pi). - Michael Somos, Sep 12 2005

Examples

			0.998136044...

References

  • K. Srinivas Rao, Ramanujan, a Mathematical Genius, Eastwest Books, Chennai Madras, 2000, p. 42.
  • Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters And Commentary, AMS, Providence RI, 1995, p. 29.
  • Bruce C. Berndt and Robert A. Rankin, Ramanujan: Essays And Surveys, AMS, Providence RI, 2001, p. 243.
  • G. H. Hardy, Ramanujan: Twelve Lectures on subjects as suggested by his Life and Work, AMS, Chelsea Providence RI, 1999, p. 8, section 1.11.

Links

Crossrefs

Cf. A019694, A091899, A158934.

Programs

  • Mathematica
    RealDigits[Exp[2*Pi/5]*(Sqrt[(Sqrt[5] + 5)/2] - GoldenRatio), 10, 100][[1]] (* Amiram Eldar, Jan 23 2022 *)
  • PARI
    {a(n)=x=exp(2/5*Pi)*(sqrt((5+sqrt(5))/2)-(1+sqrt(5))/2); floor(x*10^(n+1))%10} /* Michael Somos, Sep 12 2005 */
  • PARI
    {a(n)= x=exp(-2*Pi); x=contfracpnqn(matrix(2,oo,i,j,if(j==1,i==1,if(i==1,x,1)^(j-2)))); x=t[1,1]/t[2,1]; floor(x*10^(n+1))%10} /* Michael Somos, Sep 12 2005 */

Formula

Equals 1/A091899.
Equals exp(2*Pi/5) * A158934. - Amiram Eldar, Jan 23 2022

A356569 Sums of powers of roots of x^4 - 2*x^3 - 6*x^2 + 2*x + 1.

Original entry on oeis.org

4, 2, 16, 38, 164, 522, 1936, 6638, 23684, 82802, 292496, 1027798, 3621284, 12741562, 44862736, 157904478, 555880964, 1956721762, 6888057616, 24246779398, 85352580004, 300452999402, 1057639862416
Offset: 0

Views

Author

Greg Dresden and Ding Hao, Aug 12 2022

Keywords

Comments

The four roots of x^4 - 2*x^3 - 6*x^2 + 2*x + 1, in order from smallest to largest, are c1 = sec(17*Pi/20)/sqrt(2) - 1, c2 = sec(Pi/20)/sqrt(2) - 1 = -A158934, c3 = sec(7*Pi/20)/sqrt(2) - 1, and c4 = sec(9*Pi/20)/sqrt(2) - 1.

Examples

			a(3) = (-1.7936045...)^3 + (-0.28407904...)^3 + (0.55753652...)^3 + (3.5201470...)^3 = 38, as expected.

Links

Crossrefs

Cf. A192380, A158934 (-c2).

Programs

  • Mathematica
    Table[Sum[(Sec[k Pi/20]/Sqrt[2] - 1)^n, {k, {1, 7, 9, 17}}], {n, 0, 30}] // Round
  • PARI
    polsym(x^4 - 2*x^3 - 6*x^2 + 2*x + 1, 22) \\ Joerg Arndt, Aug 14 2022

Formula

a(n) = 2*a(n-1) + 6*a(n-2) - 2*a(n-3) - a(n-4).
G.f.: 2*(2-3*x-6*x^2+x^3)/(1-2*x-6*x^2+2*x^3+x^4).
a(n) = b(n+1) + 6*b(n) - 3*b(n-1) - 2*b(n-2) for b(n) = A192380(n).
Showing 1-2 of 2 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.