OFFSET
0,4
COMMENTS
Terms are even if and only if n is a multiple of 3. This can be shown by induction. The predictable pattern means the sequence is given by a linear recurrence with constant coefficients. - Andrew Howroyd, Dec 15 2021
LINKS
Ben Whitmore, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-5,0,0,2).
FORMULA
G.f.: x*(1 + 2*x^3)*(1 + x + 2*x^2 - x^3 + x^4)/((1 - x)^2*(1 + x + x^2)^2*(1 - 2*x^3)). - Andrew Howroyd, Dec 15 2021
From Ben Whitmore, Dec 17 2021: (Start)
a(n) = 2^(n/3+3)-2n-8 if n == 0 (mod 3);
a(n) = 2^((n+5)/3)-3 if n == 1 (mod 3);
a(n) = 3*2^((n+4)/3)-2n-7 if n == 2 (mod 3).
(End)
EXAMPLE
Start at a(0)=0.
0 is even, so to get a(1), divide by 2 and add n: a(1) = 0/2 + 1 = 1.
1 is odd, so to get a(2), add the previous term: a(2) = a(1) + a(0) = 1 + 1 = 1.
Continuing, we get
n a(n)
- ----
0 0
1 0/2 = 0 + 1 = 1
2 1 + 0 = 1
3 1 + 1 = 2
4 2/2 = 1 + 4 = 5
5 5 + 2 = 7
6 7 + 5 = 12
7 12/2 = 6 + 7 = 13
MATHEMATICA
a[0] = 0; a[n_] := a[n] = If[EvenQ[a[n - 1]], a[n - 1]/2 + n, a[n - 1] + a[n - 2]]; Array[a, 50, 0] (* Amiram Eldar, Dec 15 2021 *)
PROG
(Python)
a = [0]
[a.append(a[-1] + a[-2] if a[-1]%2 else a[-1]//2 + n) for n in range(1, 49)]
print(a) # Michael S. Branicky, Dec 15 2021
(PARI) seq(n)={my(a=vector(n)); a[1]=0; for(n=2, #a, a[n] = if(a[n-1]%2==0, a[n-1]/2+(n-1), a[n-1]+a[n-2])); a} \\ Andrew Howroyd, Dec 15 2021
(PARI) concat([0], Vec((1 + 2*x^3)*(1 + x + 2*x^2 - x^3 + x^4)/((1 - x)^2*(1 + x + x^2)^2*(1 - 2*x^3)) + O(x^30))) \\ Andrew Howroyd, Dec 15 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gavin Lupo, Dec 15 2021
STATUS
approved