A381788 Greedy expansion of Pi-3 in a base with place values 1/(10^k-1), k >= 1, using digits {0,1,2,...,8,9,A=10}.

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

Offset: 1

Author

Simon Plouffe, Mar 07 2025

Keywords

• nonn
• base

Comments

From Pontus von Brömssen, Mar 13 2025: (Start)

Since the ratio of successive place values is less than 1/10, a digit A=10 is sometimes needed. For example, if 10*A073668-37/33 < x < 1/9, the expansion of x must have an A at the second position after the radix point (for any choice of digits, not only greedy).

The expansion is not unique without specifying greedy choice of digits. For example, the number 11/1000 can be represented both as 0.010898908982... and (non-greedily) as 0.00A989899171... in this system.

For a random number, the probability that the digit A occurs decreases exponentially with the position in the expansion (with greedy choice of digits), so it seems very unlikely that 10 is a term of this sequence.

(End)

Crossrefs

Cf. A000796, A073668.

Programs

Maple

BASEN:= proc(x, b, sgn, k)
local i, j, v, premier, fin, lll, liste, w, baz;
    baz := evalf(b);
    v := abs(frac(evalf(x)));
    fin := trunc(evalf(Digits/log10(b)));
    lll := [seq(i^k*(baz^i + sgn), i = 1 .. fin)];
    liste := [];
    for i to fin do w := trunc(v*lll[i]); v := v - w/lll[i]; liste := [op(liste), w] end do;
    RETURN(liste)
end;
BASEN(Pi-3,10,-1,0);

Formula

Sum_{k>=1} a(k)/(10^k - 1) = Pi - 3.

Extensions

Edited by N. J. A. Sloane, Mar 18 2025