Fibonacci Accumulation Sequence

[Davide Rotondo]

https://oeis.org/A163227
Consider the sequence A163227
Fibonacci Accumulation Sequence.
1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 8, 20, 13, 33, 21, 54, 34, 88, 55, 143, 89, 232, 144, 376, 233, 609, 377, 986, 610, 1596, 987, 2583, 1597, 4180, 2584, 6764, 4181, 10945, 6765, 17710, 10946, 28656, 17711, 46367, 28657, 75024, 46368, 121392, 75025, 196417
It is known that each element is given by the formula a(n) = 2*a(n-2) - a(n-6), where a(1,2,3)=1, that is, each value is equal to the sum of the second preceding element multiplied by 2 plus the sixth preceding element...
I also noticed that you can use another formula that consists of a double formula for values with even or odd indexes...
For an even index, each value is given by the sum of the first preceding element plus the second preceding element... a(n) = a(n-1) + a(n-2); for an odd index, each value is given by the sum of the second preceding element plus the fourth preceding element... a(n) = a(-2) + a(n-4).

What do you think?

See you soon

Davide

[Kevin Ryde]

Davide Rotondo <david...@gmail.com> writes:
>
> A163227 Fibonacci Accumulation Sequence.

Ah, yeah, that's Fibonacci(x) interleaved with Fibonacci(y)-1.
They're shifted relative to each other to stop that being obvious.

> odd index, ... a(n) = a(-2) + a(n-4)

Yes that's the Fibonacci recurrence.

(And yes, all of the program code should have used a Fibonacci
function instead of ad-hoc length initial from the gf.)