cp's OEIS Frontend

From Samuel Harkness, Mar 02 2023: (Start)

This sequence is partly defined by the following cases for a(353) onwards. But what is the pattern to the 1s and 2s?

For k >= 0:

a(353+4k) = 1 or 2;

a(354+2k) = (354+2k)/2 - 63;

a(355+4k) = 1.

Of the 14912 n equal to 1 (mod 4) between 352 and 60000, 4983 (33.416%) equal 1 and 9929 (66.584%) equal 2. From observation it appears that 1/3 of a(353+4k); k>=0 equal 1 and 2/3 equal 2 (see Figure).

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From Samuel Harkness, Mar 11 2023: (Start)

Observations:

Record values k > 2 seem to occur at the following places:

First k for k == 0 (mod 3) occurs at n = 2^(k/3+2) + k/3 - 4;

First k for k == 1 (mod 3) occurs at n = 2^((k-1)/3+2) + (k-1)/3 - 3;

First k for k == 2 (mod 3) occurs at n = 3*(2^((k+1)/3)) + (k-14)/3.

For any value k, frequency(k) ~= 2*frequency(3+k). For any value j >= 0, frequency(2+j) ~= frequency(3+j) ~= frequency(4+j).

This sequence contains many recurring strings. For example, {1, 3, 4, 1, 2} occurs 12499 times in the first 100000 terms. From its 5th occurrence at a(40) through its 64th occurrence at a(517), the number of terms between each {1, 3, 4, 1, 2} minus one gives

{1 3 1 4 ...}

First 1 term of A001511, 3, first 1 term of A001511, 4.

{... 1 2 1 4 1 2 1 5 ...}

First 3 terms of A001511, 4, first 3 terms of A001511, 5.

{... 1 2 1 3 1 2 1 5 1 2 1 3 1 2 1 6 ...}

First 7 terms of A001511, 5, first 7 terms of A001511, 6.

{... 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 7 ...}

First 15 terms of A001511, 6, first 15 terms of A001511, 7.

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