A076539 -id:A076539 - cp's OEIS frontend

A144609 Sturmian word of slope Pi.

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1

Offset: 0

N. J. A. Sloane, Jan 13 2009

nonn

A063438 seems to contain the run lengths of 1's. - R. J. Mathar, May 30 2025

See A144595 for further details.
Seems to be very similar to A070127. Is this a coincidence?
Cf. A063438, A076539 (partial sums).

Digits := 500 :
x :=1 ;
y :=0 ;
slop := Pi ;
printf("0,") ;
for n from 1 to 300 do
    if evalf((y+1)/x-slop) > 0 then
        x := x+1 ;
        printf("0,") ;
    else
        y := y+1 ;
        printf("1,") ;
    end if;
end do: # R. J. Mathar, May 30 2025

christoffel[s_, M_] := Module[{n, x = 1, y = 0, ans = {0}}, Do[ If[y + 1 <= s*x, AppendTo[ans, 1]; y++, AppendTo[ans, 0]; x++], {n, 1, M}]; ans]; christoffel[Pi, 105] (* Robert G. Wilson v, Feb 02 2017, after Jean-François Alcover, Sep 19 2016, A274170 *)

A093700 Number of 9's immediately following the decimal point in the expansion of (3+sqrt(8))^n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 55, 56, 57

Offset: 1

Marvin Ray Burns, Apr 10 2004

Number of 0's immediately following the decimal point in the expansion of (3-sqrt(8))^n.

			Let n=10, (3+sqrt(8))^10= 45239073.9999999778... (the fractional part starts with seven 9's), so the 10th element in this sequence is 7.
The 132nd element is 100. The 1000th element is 765. The 1307th element is 1000.
The arrangement of repeating elements are like A074184 (Index of the smallest power of n >= n!) and A076539 (Numerators a(n) of fractions slowly converging to pi) and A080686 (Number of 19-smooth numbers <= n).

Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Experiment. Math. Volume 9, Issue 1 (2000), 3-12, Project Euclid - Cornell Univ (see Proposition 1).
Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Math Forum, Triangular Numbers That are Perfect Squares
Math Pages, On m = sqrt(sqrt(n) + sqrt(kn+1)) [Wrong link]
Robert Simms, Using counting numbers to generate Pythagorean triples

Cf. A003499, A050608, A080686, A076539, A074184.

For[n = 1, n < 999, n++, Block[{$MaxExtraPrecision = 50*n}, Print[ -Floor[Log[10, 1 - N[FractionalPart[(3 + 2Sqrt[2])^n], n]]] - 1]]]
f[n_] := Block[{}, -MantissaExponent[(3 - Sqrt[8])^n][[2]]]; Table[ f[n], {n, 75}] (* Robert G. Wilson v, Apr 10 2004 *)

Roughly, floor(3*n/4)

cp's OEIS Frontend

A144609 Sturmian word of slope Pi.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica