A245645 Decimal expansion of the common value of A and B in Daniel Shanks' "incredible identity" A = B.

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

Offset

1,1

Comments

See the Spohn reference for a generalization of this equality. - Joerg Arndt, May 01 2016

References

Henri Cohen, A Course in Computational Algebraic Number Theory, 3., corr. print., Springer-Verlag Berlin Heidelberg New York, 1996, Exercise 7 for Chapter 4, page 218.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Daniel Shanks, Incredible Identities, Part 1, Part 2, The Fibonacci Quarterly 12, no. 3 (1974):271, 280.

William G. Spohn, Jr., Letter to the editor, The Fibonacci Quarterly 14, no. 1 (1976).

Robert Allen Supencheck, Some examples in low-dimensional algebraic number theory, 1990; see chapter 1.

Formula

Equals A = sqrt(5) + sqrt(22 + 2*sqrt(5)),

Equals B = sqrt(11 + 2*sqrt(29)) + sqrt(16 - 2*sqrt(29) + 2*sqrt(55 - 10*sqrt(29))).

The minimal polynomial of both A and B is x^4 - 54*x^2 - 40*x + 269.

Example

7.3811759408956579709872668754651303326656461102953475776191021866181514...

Mathematica

RealDigits[Sqrt[5] + Sqrt[22 + 2*Sqrt[5]], 10, 101] // First

Crossrefs

Sequence in context: A248281 A111197 A372778 * A335994 A115411 A019726

Adjacent sequences: A245642 A245643 A245644 * A245646 A245647 A245648

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jul 28 2014

Status

approved