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Several subsets of this sequence can be defined, each of them proving that the there exists an infinite number of seeds in the Reverse and Add! procedure in base 2:

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From A.H.M. Smeets, May 30 2019: (start)

Several subsets of this sequence can be defined, each of them proving that the there exists an infinite number of seeds in the Reverse and Add! procedure:

In the string representations, + stands for concatenation and ^ stands for repeated concatenation.

Example 1:

With f_1(n) = 81*2^n + 12*2^floor(n/2) - 60, {f_1(n) | n >= 11} is a proper subset.

String representation for the binary representation of f_1(n): 1010001 + (0)^floor((n-11)/2 + 0010 + (1)^floor((n-10)/2) + 1000100 for n >= 11.

f_1(n) = 3*f_1(n-1) - 6*f_1(n-3) + 4*f_1(n-4) for n > 3, f_1(0) = 33, f_1(1) = 114, f_1(2) = 288, f_1(3) = 612 (from Colin Barker).

f_1(n) = (-60 + 81*2^n + 3*2^((1+n)/2)*(1+(-1)^(n+1) + sqrt(2) + (-1)^n*sqrt(2))) (from Colin Barker).

G.f. for f_1: 3*(11 + 5*x - 18*x^2 - 18*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)) (from Colin Barker).

Example 2:

With f_2(n) = 32*8^n + 64*(8^n - 8)/7 + 68, {f_2(n) | n >= 2} is a proper subset.

String representation for the binary representation of f_2(n): 10 + (100)^(n-2) + 1000100 for n >= 2.

f_2(n) = 8*f_2(n-1) + 36 for n > 0, a(0) = 0.

f_2(n) = 36*(-1 + 8^(n+1))/7 (from Colin Barker).

f_2(n) = 9*f_2(n-1) - 8*f_2(n-2) for n > 1, a(0) = 0, a(1) = 36 (from Colin Barker).

G.F. for f_2: 36 / ((1 - x)*(1 - 8*x)) (from Colin Barker).

Example 3:

f_3(n) = 369*2^n - 24*2^floor(n/2) +132, {f_3(n) | n >= 12} is a proper subset.

String representation for the binary representation of f_3(n): 101110000 + (1)^ceiling((n-12)/2) + 101 + (0)^floor((n-12)/2) + 010000100 for n >= 12.

Example 4:

f_4(n) = 21*2^n + 6*2^floor(n/2) - 12, {f_4(n) | n >= 12} is a proper subset.

String representation for the binary representation of f_4(n): 10101 + (0)^floor((n-9)/2) + 0010 + (1)^ceiling((n-9)/2) + 10100 for n >= 12.

f_4(n) = 2*f_4(n-1) + 12*(-1)^n for n >= 4. (End)

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A.H.M. Smeets, <a href="/A306516/b306516.txt">Table of n, a(n) for n = 1..1701</a>

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