|
%I #15 Dec 15 2015 12:50:34
%S 3,9,3,1,8,8,4,7,7,0,4,9,6,4,4,3,2,4,4,9,7,2,5,8,2,1,3,1,3,8,9,0,3,8,
%T 8,5,8,5,4,8,3,9,1,4,0,7,8,8,6,6,2,8,6,9,5,3,9,2,9,3,2,4,7,5,7,5,7,8,
%U 7,7,5,8,3,3,8,9,7,4,9,8,6,6,8,1,0,9,7,6,6,6,2,0,6,1,0,1,8,5,8,8,8,0,1,3,3,3,0,0,8,0,5,9,3,2,2,6,3,1,5,3,2,6,8,0,9,0,4,7,5,0,4,9,4,2,6,6,6,1,2,1,1,4,2,4,3,3,4,9,8,4,4,3,5,8,4,7,7,5,8,5,0,6,5,5,9,3,3,7,2,5,0,9,1,4,3,2,8,8,7,7,0,5,4,3,2,2,3,1,4,0,7,7,1,7,1,7,5,9,5,3,3,3,7,7,6
%N Decimal expansion of constant z = Sum_{n>=1} {(3/2)^n} / 2^n, where {x} denotes the fractional part of x.
%H Paul D. Hanna, <a href="/A264918/b264918.txt">Table of n, a(n) for n = 1..500</a>
%F z = Sum_{n>=1} (3^n mod 2^n) / 4^n = Sum_{n>=1} A002380(n) / 4^n.
%F 3 - z = Sum_{n>=1} [(3/2)^n] / 2^n = Sum_{n>=1} A002379(n) / 2^n, where [x] denotes the integer floor function of x.
%e z = 0.39318847704964432449725821313890388585483914078866\
%e 28695392932475757877583389749866810976662061018588\
%e 80133300805932263153268090475049426661211424334984\
%e 43584775850655933725091432887705432231407717175953\
%e 33776901692614854937460993931094741172922114373160\
%e 19617637538747813543456758934332723336245738884968...
%e INFINITE SERIES.
%e (1) z = 1/4 + 1/4^2 + 3/4^3 + 1/4^4 + 19/4^5 + 25/4^6 + 11/4^8 + 161/4^9 + 227/4^10 + 681/4^11 + 1019/4^12 +...+ A002380(n)/4^n +...
%e (2) 3 - z = 1/2 + 2/2^2 + 3/2^3 + 5/2^4 + 7/2^5 + 11/2^6 + 17/2^7 + 25/2^8 + 38/2^9 + 57/2^10 + 86/2^11 + 129/2^12 + 194/2^13 + 291/2^14 +...+ A002379(n)/2^n +...
%e where
%e 3 - z = 2.60681152295035567550274178686109611414516...
%Y Cf. A002379 ([(3/2)^n]), A002380 (3^n mod 2^n), A264919, A264920, A264921, A264922.
%K nonn,cons
%O 1,1
%A _Paul D. Hanna_, Dec 03 2015
|
