cp's OEIS Frontend

For all n, a(A000079(n)) = A000225(n+1), i.e. a(2^n) = (2^(n+1))-1.

For n>=1, a(A000225(n)) = A000325(n).

This permutation is obtained by "entangling" even and odd numbers with complementary pair A005187 & A055938, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A005187(n+1) to the parent node containing n, and each child to the right is obtained as A055938(n):

0

|

...................1...................

3 2

7......../ \........6 4......../ \........5

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

15 14 11 13 8 9 10 12

31 30 26 29 22 24 25 28 16 17 18 20 19 21 23 27

etc.

For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to 0 from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.

Permutation A233278 gives the mirror image of the same tree.

This permutation is obtained by "entangling" even and odd numbers with complementary pair A055938 & A005187, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A055938(n) to the parent node containing n, and each child to the right is obtained as A005187(n+1):

0

|

...................1...................

2 3

5......../ \........4 6......../ \........7

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

12 10 9 8 13 11 14 15

27 23 21 19 20 18 17 16 28 25 24 22 29 26 30 31

etc.

For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to zero at the root from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.

Permutation A233276 gives the mirror image of the same tree.

Equally: One-based row index of n in array A255557.

In other words, if n = A005187(k) for some k >= 1, then a(n) = k-1, otherwise it must be that n = A055938(h) for some h, and then a(n) = h.

In binary trees like A233276 and A233278, a(n) gives the contents at the parent node of node containing n, for any n >= 1.

When iterating a(n), a(a(n)), a(a(a(n))), and so on, A070939(n) = A256478(n) + A256479(n) = A257248(n) + A257249(n) gives the number of steps needed to reach zero, from any starting value n >= 1.

a(n) tells how many terms of A055938 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n. This count includes also n in case it itself is a term of A055938. See also comments in A256478 and A256991.

a(n) tells how many nonzero terms of A005187 are encountered when traversing toward the root of binary tree A233276, starting from the node containing n and before 1 is reached. This count includes both n (in case it is a term of A005187) but excludes the 1 and 0 at the root. See also comments in A257249, A256478 and A256991.