A190287 Decimal expansion of (5+sqrt(25+4*sqrt(5)))/2.

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

Offset

1,1

Comments

The rectangle R whose shape (i.e., length/width) is (5+sqrt(25+4r))/2, where r=sqrt(5), can be partitioned into rectangles of shapes 5 and r in a manner that matches the periodic continued fraction [5, r, 5, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [5,2,2,2,1,1,1,10,1,1,2,1,...] at A190288. For details, see A188635.

Links

Table of n, a(n) for n=1..120.

Index entries for algebraic numbers, degree 4.

Formula

Minimal polynomial: x^4 - 10*x^3 + 25*x^2 - 5. - Amiram Eldar, May 28 2026

Example

5.413085645411028710287065567557494135315932736504125841550513375922677449233....

Mathematica

r=5^(1/2)
FromContinuedFraction[{5, r, {5, r}}]
FullSimplify[%]
RealDigits[N[%%, 120]]
N[%%%, 40]

Prog

(PARI) (5+sqrt(25+4*sqrt(5)))/2 \\ Charles R Greathouse IV, May 18 2026

Crossrefs

Cf. A188635, A190288.

Sequence in context: A268911 A351123 A166044 * A087707 A390735 A386744

Adjacent sequences: A190284 A190285 A190286 * A190288 A190289 A190290

Keyword

nonn,cons,changed

Author

Clark Kimberling, May 07 2011

Status

approved