A071873 - OEIS

%I #28 May 13 2015 22:04:21

%S 5,0,5,0,5,0,5,0,5,0,5,0,5,0,5,0,5,0,5,1,6,1,6,1,6,1,6,1,6,1,6,1,6,1,

%T 6,1,6,1,6,2,7,2,7,2,7,2,7,2,7,2,7,2,7,2,7,2,7,2,7,3,8,3,8,3,8,3,8,3,

%U 8,3,8,3,8,3,8,3,8,3,8,4,9,4,9,4,9,4,9,4,9,4,9,4,9,4,9,4,9,4,0,5

%N Decimal expansion of the sixth (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.

%C In other words, this constant satisfies x = Sum_{n>=0} ( floor(10*n*x) (mod 10) ) / 10^n.

%C The sixth selvage number is equal to the complement of the fifth selvage number: s_6 = 1 - s_5.

%H MathWorld, <a href="http://mathworld.wolfram.com/EquidistributedSequence.html">Equidistributed Sequence</a>

%F a(n) = floor[10*(n*x)] (Mod 10), where x = sum{k=1..inf} a(k)/10^k.

%F a(n) = 9 - A071793(n).

%e x=0.50505050505050505051616161616161616161627272727272...

%e a(7) = 5 since floor(10*(7*x)) (Mod 10) = 5.

%e The multiples of this constant x begin:

%e 1*x = 0.5050505050505050505161616161616161616163...

%e 2*x = 1.010101010101010101032323232323232323233...

%e 3*x = 1.515151515151515151548484848484848484849...

%e 4*x = 2.020202020202020202064646464646464646465...

%e 5*x = 2.525252525252525252580808080808080808081...

%e 6*x = 3.030303030303030303096969696969696969698...

%e 7*x = 3.535353535353535353613131313131313131314...

%e 8*x = 4.040404040404040404129292929292929292930...

%e 9*x = 4.545454545454545454645454545454545454546...

%e 10*x = 5.050505050505050505161616161616161616163...

%e 11*x = 5.555555555555555555677777777777777777779...

%e 12*x = 6.060606060606060606193939393939393939395...

%e wherein the tenths place of n*x yields the n-th digit of x.

%t k = 5; f[x_] := Floor[10*FractionalPart[x]]; Clear[xx]; xx[n_] := xx[n] = Catch[ For[x = xx[n - 1], True, x += 10^(-n), If[f[n*x] == f[10^(n - 1)*x], Throw[x]]]]; xx[1] = k/10; Scan[xx, Range[100]]; RealDigits[xx[100]][[1]] (* _Jean-François Alcover_, Dec 06 2012 *)

%t Clear[a]; a[1] = 5; a[2] = 0; a[n0=3] = 5; a[_] = 0; digits = 10^(n0-1); Do[a[n] = Mod[Floor[10*n*Sum[a[k]/10^k, {k, 1, n}]], 10], {n, n0+1, digits}]; Table[a[n], {n, 1, digits}] (* _Jean-François Alcover_, May 12 2015 *)

%Y Cf. A071789, A071790, A071791, A071792, A071793, A071874, A071875, A071876, A071877.

%Y Cf. A257875 (continued fraction).

%K cons,easy,nonn,base

%O 0,1

%A _Paul D. Hanna_, Jun 10 2002