1,2

Remainders for classes m of integers n (mod 2^(m+1)). After applying one Collatz (3x+1)-transformation to the so-classified integers the result can be written in two classes (mod 6) only.

This classifying scheme covers all integers >0.

With one 3x+1-transformation T(x;p) := x'= (3x+1)/2^p

all numbers x, described in the form, with the free parameter i>=0, x = i*2^N +a(N) result in x', describable by the two classes with the same parameter i:

x' = i*6 + 1 (for odd N>2), or x' = i*6 + 5 (for even N). Thus

x = 4*i + 3 -> x' = 6*i + 5, x = 8*i + 1 -> x' = 6*i + 1,

x =16*i +13 -> x' = 6*i + 5, x = 32*i + 5 -> x' = 6*i + 1,

x =64*i +53 -> x' = 6*i + 5, x =128*i +21 -> x' = 6*i + 1,

....

all with "i" as a free parameter >=0 covering all natural numbers

a(2m) = (5*2^(2m-1) - 1)/3, a(2m-1) = (2^(2m-2)-1)/3.

a(2n+1)=10a(2n)+3. a(n+1)-2a(n)= A001045(n+3), signed. - Paul Curtz (bpcrtz(AT)free.fr), Jul 01 2008

a(1) = (2^0-1)/3 = 0, a(2) = (5*2^1 - 1) / 3 = 3,

a(3) = (2^2-1)/3 = 1, a(4) = (5*2^3 - 1) / 3 =13,

a(5) = (2^4-1)/3 = 5, a(6) = (5*2^5 - 1) / 3 =53,

a(7) = (2^6-1)/3 =21.

....

a[1] = 0; a[2] = 3; a[n_] := a[n] = 4a[n - 2] + 1; Table[ a[n], {n, 35}] (from Robert G. Wilson v Aug 20 2004)

Bisections are A002450 & A072197.

Sequence in context: A016479 A134768 A113139 this_sequence A118384 A133176 A089435

Adjacent sequences: A096770 A096771 A096772 this_sequence A096774 A096775 A096776

easy,nonn,new

Gottfried Helms (helms(AT)uni-kassel.de), Aug 15 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 20 2004