A134944 Decimal expansion of (1 + sqrt(5))/8, the golden ratio divided by 4.

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

Offset

0,1

Comments

Area of quadrilateral with lengths (sqrt(5)+1)/4, (sqrt(5)-1)/4, 1/2, 1. This quadrilateral can have a distance of 1 or phi when Brahmagupta's Formula is used.

((sqrt(5)+1)/4)^2+((sqrt(5)-1)/4)^2+(1/2)^2=1. Inverses of three consecutive squared fibonacci numbers when normalized will approach the three values as the limit goes to infinity. - Thomas Olson, Sep 07 2014

Links

Table of n, a(n) for n=0..85.

Formula

Equals sqrt((sqrt(5)+3)/4*((sqrt(5)+3)/4-(sqrt(5)+1)/4)*((sqrt(5)+3)/4-(sqrt(5)-1)/4)*((sqrt(5)+3)/4-1/2)*((sqrt(5)+3)/4-1)). - Thomas Olson, Sep 07 2014

F(n) is a fibonacci number

lim_n->inf {(-F(n)^2+F(n+1)^2+F(n+2)^2)/(F(n)^2+F(n+1)^2+F(n+2)^2)} = A019863. - Thomas Olson, Sep 16 2014

lim_n->inf {(F(n)^2-F(n+1)^2+F(n+2)^2)/(F(n)^2+F(n+1)^2+F(n+2)^2)}=1/2. - Thomas Olson, Sep 16 2014

lim_n->inf {(-F(n)^2-F(n+1)^2+F(n+2)^2)/(F(n)^2+F(n+1)^2+F(n+2)^2)}= A019827. - Thomas Olson, Sep 16 2014

Example

0.404508497187...

Maple

Digits:=100: evalf((1+sqrt(5))/8); # Wesley Ivan Hurt, Sep 08 2014

Mathematica

RealDigits[(1 + Sqrt[5])/8, 10, 100][[1]] (* Wesley Ivan Hurt, Sep 08 2014 *)

Prog

(PARI) a(n)=floor(10^n*(1+sqrt(5))/8)%10 \\ Edward Jiang, Sep 07 2014

Crossrefs

Cf. A001622, A019863.

Sequence in context: A258849 A258859 A258860 * A088375 A217220 A361620

Adjacent sequences: A134941 A134942 A134943 * A134945 A134946 A134947

Keyword

cons,nonn,easy

Author

Omar E. Pol, Nov 15 2007

Status

approved