A318525 Decimal expansion of ((3+2*5^(1/4))/(3-2*5^(1/4)))^(1/4).

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

Offset

1,1

Comments

Ramanujan's question 1070 (iii) asks for a proof of the identity

((3+2*5^(1/4))/(3-2*5^(1/4)))^(1/4) = (5^(1/4)+1)/(5^(1/4)-1).

This is the larger of the two real roots of x^4 - 6*x^3 + 6*x^2 - 6*x + 1 = 0.

References

Srinivasa Ramanujan, Collected Papers, Chelsea, 1962, page 334, Question 1070.

Links

Table of n, a(n) for n=1..86.

Example

5.0375591418015601791686190145827146563721270377443099468187040056...

Mathematica

RealDigits[Surd[(3 + 2*5^(1/4))/(3 - 2*5^(1/4)), 4], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

Prog

(PARI) ((3+2*5^(1/4))/(3-2*5^(1/4)))^(1/4)
(PARI) p(x)=x^4-6*x^3+6*x^2-6*x+1;
solve(x=5, 6, p(x)) \\ Hugo Pfoertner, Sep 12 2018

Crossrefs

Cf. A318523, A318524.

Sequence in context: A019947 A193182 A139399 * A021669 A011511 A020856

Adjacent sequences: A318522 A318523 A318524 * A318526 A318527 A318528

Keyword

nonn,cons

Author

Hugo Pfoertner, Aug 28 2018

Status

approved