cp's OEIS Frontend

Records appear to be given by A001045 Jacobsthal numbers.

(a(n)-1)/2 appears to be A085358.

The terms between the A001045(n+3) are:

3

5

11

3,

21

3, 5,

43

3, 5, 11, 3,

85

3, 5, 11, 3, 21, 3, 5,

171

3, 5, 11, 3, 21, 3, 5, 43, 3, 5, 11, 3,

341

3, 5, 11, 3, 21, 3, 5, 43, 3, 5, 11, 3, 85, 3, 5, 11, 3, 21, 3, 5,

683

This gives the same sequence. Every column has the same number.

By rows, there are 0, 0, 1, 2, 4, 7, 12, 20, ... apparently = Fibonacci(n+1) - 1 = A000071 terms.

From Paul Curtz, Jan 26 2016: (Start)

a(n) is also in

0, 1, 1 0, 3, 0, 1, 5, 0, ... equivalent to A035614(n)

1, 1, 3, 1, 5, 1, 1, 11, 1, ... equivalent to A035612(n)

1, 3, 5, 1, 11, 1, 3, 21, 1, ... (compare to A268032)

3, 5, 11, 3, 21, 3, 5, 43, 3, ... a(n) (equivalent to a3(n) in A035612)

5, 11, 21, 5, 43, 5, 11, 85, 5, ...

etc.

Every vertical comes from A001045 (*).

Second row: first one removing all 0's.

Third row: second one removing a part of 1's respecting (*)

Fourth row: third one removing all 1's.

etc.

The offset 0 is homogeneous to these sequences. (End)

About c-equivalent see in comment in A233249.

a(n) is even iff A171791(n+1) is odd - holds for at least the first 1028 terms. The reason, put very briefly, is that: a(n) is even if and only if n is the double of a "fibbinary number". Cf. A267508. [Jörgen Backelin, Jan 15 2016 added by Jeremy Gardiner, Jan 26 2016]