A223140 Decimal expansion of (sqrt(29) + 1)/2.

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

Offset

1,1

Comments

Decimal expansion of sqrt(7 + sqrt(7 + sqrt(7 + sqrt(7 + ... )))).

Sequence with a(1) = 2 is decimal expansion of sqrt(7 - sqrt(7 - sqrt(7 - sqrt(7 - ... )))) - A223141.

From Wolfdieter Lang, Jan 05 2024: (Start)

This number phi29 = (1 + sqrt(29))/2 is the fundamental algebraic integer in the quadratic number field Q(sqrt(29)) with minimal polynomial x^2 - x - 7. The other root is -A223141.

phi29^n = 7*A(n-1) + A(n)*phi29, where A(n) = A015442(n) with A(-1) = 1/7, for n >= 0. For negative powers n see A367454 = 1/phi29. (End)

Links

Table of n, a(n) for n=1..102.

Index entries for algebraic numbers, degree 2.

Formula

Closed form: (sqrt(29) + 1)/2 = A098318-2 = 10*A085551+3 = A223141+1.

sqrt(7 + sqrt(7 + sqrt(7 + sqrt(7 + ... )))) - 1 = sqrt(7 - sqrt(7 - sqrt(7 - sqrt(7 - ... )))). See A223141.

Example

3.1925824035672520156253552457701...

Mathematica

RealDigits[(1 + Sqrt[29])/2, 10, 130]

Prog

(PARI) (sqrt(29)+1)/2 \\ Charles R Greathouse IV, Feb 10 2025

Crossrefs

Cf. A223141, A367454.

Essentially the same as A098318 and A085551.

Sequence in context: A157674 A360752 A063467 * A021762 A019736 A213595

Adjacent sequences: A223137 A223138 A223139 * A223141 A223142 A223143

Keyword

nonn,cons

Author

Jaroslav Krizek, Apr 02 2013

Status

approved