A341312 a(n) = a(n-1) + a(n-3) unless a(n-1) and a(n-3) are both even in which case a(n) = (a(n-1) + a(n-3))/2, with a(0) = a(1) = a(2) = 1.

1, 1, 1, 2, 3, 4, 3, 6, 5, 8, 7, 12, 10, 17, 29, 39, 56, 85, 124, 90, 175, 299, 389, 564, 863, 1252, 908, 1771, 3023, 3931, 5702, 8725, 12656, 9179, 17904, 15280, 24459, 42363, 57643, 82102, 124465, 182108, 132105, 256570, 219339, 351444, 304007, 523346, 437395, 741402, 632374

Offset: 0

Author

N. J. A. Sloane, Feb 16 2021

Keywords

• nonn

Comments

A sequence intermediate between Narayana's A000930 and Reed Kelly's A214551.

It will be interesting to compare the growth rates of A000930 (well-understood), A241551 (a mystery), the present sequence, and A341313.

It appears that the equation log(a(n)) = 0.296869*n - 4.69131 is a good fit to the data (see the figures). - Hugo Pfoertner, Feb 17 2021

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..5000
• Hugo Pfoertner, Comparison of linear fits to logarithm of A341312, A341313, A214551 (Reed Kelly), and A000930 (Narayana's cows).
• Hugo Pfoertner, Deviation of log(A341312) from linear fit in range 3...10000.

Crossrefs

Cf. A000930, A214551, A341313.

Programs

Maple

RK2:=proc(n) local t1; option remember;
if n <= 2 then 1 else t1:=RK2(n-3)+RK2(n-1);
   if (RK2(n-3) mod 2) = 0 and (RK2(n-1) mod 2) = 0 then t1:=t1/2; fi;
t1; fi; end;
[seq(RK2(n),n=0..60)];

PARI

a341312(nterms)={my(a=vector(nterms));a[1]=a[2]=1;a[3]=2;for(n=4,nterms,a[n]=if(a[n-1]%2==0&&a[n-3]%2==0,(a[n-1]+a[n-3])/2,a[n-1]+a[n-3]));concat([1],a)};
a341312(60) \\ Hugo Pfoertner, Feb 17 2021