A259656 -id:A259656 - cp's OEIS frontend

0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1

Offset: 0

Allan C. Wechsler, Apr 25 2018

The definition and early trajectory are strikingly reminiscent of A259656. The first difference between the two sequences is at n = 38, where A259656 has 4 and this sequence has 2.
Start with n, and repeatedly subtract the largest possible binary palindrome that leaves a nonnegative residue. This sequence gives the number of such steps needed to reach 0.
This sequence is unbounded, since the gaps between binary palindromes are arbitrarily large, but it grows very slowly.
If we search for the smallest partition into binary palindromes, not necessarily greedy, we get another sequence dominated by this one. The first difference is at n = 44. It is believed that this smaller sequence is bounded, but I have not been able to find a claim of the maximum. See Cilleruelo and Luca 2016.
The position where n = 0.. occurs for the first time: 0, 1, 2, 11, 44, 557, 131630, ... - Antti Karttunen and Altug Alkan, May 13 2018

			For n = 44:
The largest palindrome not exceeding 44 is 33 (100001). 44 - 33 = 11.
The largest palindrome not exceeding 11 is 9 (1001). 11 - 9 = 2.
The largest palindrome not exceeding 2 is 1. 2 - 1 = 1.
The largest palindrome not exceeding 1 is 1. 1 - 1 = 0.
The iteration took 4 steps to reach 0, so a(44) = 4.
For n = 131630; A303534(131630) = 557 and A303534(557) = 44. Since a(44) = 4 (as above), a(557) = 5 and a(131630) = 6. - _Altug Alkan_, Apr 26 2018

Antti Karttunen, Table of n, a(n) for n = 0..65537
Javier Cilleruelo and Florian Luca, Every Positive Integer is a Sum of Three Palindromes.
Index entries for sequences related to binary expansion of n

Cf. A006995 (binary palindromes), A206913, A259656, A303534.

isA006995(n) = Vecrev(n=binary(n))==n;
A303534(n) = {my(k=0); while(!isA006995(n-k), k++); k; } \\ From A303534
A303536(n) = if(!n,n,1+A303536(A303534(n))); \\ Antti Karttunen, May 13 2018

a(0) = 0; for n > 0, a(n) = 1 + a(A303534(n)). [We are iterating the map n -> A303534(n) until zero is reached.] - Antti Karttunen, May 13 2018, after an earlier comment.

More terms from Altug Alkan, Apr 25 2018

cp's OEIS Frontend

A303536 Number of terms in greedy partition of n into binary palindromes.

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