A340709 Let k = n/2 + floor(n/4) if n is even, otherwise (3n+1)/2; then a(n) = A093545(k).

0, 1, 2, 3, 5, 4, 7, 6, 10, 8, 12, 9, 15, 11, 17, 13, 20, 14, 22, 16, 25, 18, 27, 19, 30, 21, 32, 23, 35, 24, 37, 26, 40, 28, 42, 29, 45, 31, 47, 33, 50, 34, 52, 36, 55, 38, 57, 39, 60, 41, 62, 43, 65, 44, 67, 46, 70, 48, 72, 49, 75, 51, 77, 53, 80, 54, 82, 56, 85, 58, 87

Offset: 0

Author

Thomas Scheuerle, Jan 16 2021

Keywords

• nonn
• easy

Comments

This is a permutations of the nonnegative integers.

A093545 is the inverse of A340615.

Some of the cycles of this permutation are: (0),(1),(2),(3),(5 4),(7 6),(10 12 15 13 11 9 8),(17 14),(20 25 21 18 22 27 23 19 16),... .

A340615 and A342131 are permutations, constructed by a small modification of Collatz function (A014682). This sequence relates these permutations which each other: A340615(a(n)) = A342131(n).

Links

• Index entries for sequences that are permutations of the nonnegative integers.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,1,0,0,0,-1).
• Index entries for sequences related to 3x+1 (or Collatz) problem.

Crossrefs

Cf. A014682, A340615, A093545, A342131.

Programs

MATLAB

function a = A340709(max_n)
    for n = 1:max_n*10
        k = (n-1)+floor(((n-1)+1)/5);
        m = n-1;
        if floor(k/2) == k/2
            A340615(n) = k/2;
        else
            A340615(n) = (k*3+1)/2;
        end
        if floor(m/2) == m/2
            b(n) = m/2+floor(m/4);
        else
            b(n) = (m*3+1)/2;
        end
    end
    for n = 1:(length(A340615)/10)
        a(n) = find(A340615==b(n))-1;
    end
end

Formula

a(4*m) = 5*m.

a(2+4*m) = 2+5*m.

a(1+6*m) = 1+5*m.

a(3+6*m) = 3+5*m.

a(4+6*m) = 4+5*m.

a(n) = -2*a(n-1) - 3*a(n-2) - 4*a(n-3) - 4*a(n-4) - 4*a(n-5) - 3*a(n-6) - 2*a(n-7) - a(n-8) + 25n - 101 for n >= 8.

a(n) = A093545(A342131(n)).

G.f.: x*(1 + 2*x + 3*x^2 + 5*x^3 + 3*x^4 + 5*x^5 + 2*x^6 + 3*x^7 + x^8)/(1 - x^4 - x^6 + x^10). - Stefano Spezia, Mar 01 2021