A190290 - OEIS

A190290

Decimal expansion of (3+sqrt(21))/3.

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

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OFFSET

1,1

COMMENTS

The rectangle R whose shape (i.e., length/width) is (3+sqrt(21))/3, can be partitioned into rectangles of shapes 3/2 and 2 in a manner that matches the periodic continued fraction [2, 3/2, 2, 3/2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [2,1,1,8,1,1,2,1,1,8,1,1,2,,...]. For details, see A188635.

LINKS

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

FORMULA

Equals 1 + Sum_{k>=0} binomial(2*k,k)/7^k. - Amiram Eldar, Aug 03 2020

EXAMPLE

2.527525231651946668862682397909336162995...

MATHEMATICA

FromContinuedFraction[{2, 3/2, {2, 3/2}}]

ContinuedFraction[%, 100] (* [2, 1, 1, 8, 1, 1, 2, ... *)

RealDigits[N[%%, 120]] (* A190290 *)

N[%%%, 40]

CROSSREFS