A341313 a(n) = (a(n-1) + a(n-3))/2^m, where 2^m is the highest power of 2 that divides both a(n-1) and a(n-3), with a(0) = a(1) = a(2) = 1.

1, 1, 1, 2, 3, 4, 3, 6, 5, 8, 7, 12, 5, 12, 6, 11, 23, 29, 40, 63, 92, 33, 96, 47, 80, 11, 58, 69, 80, 69, 138, 109, 178, 158, 267, 445, 603, 870, 1315, 1918, 1394, 2709, 4627, 6021, 8730, 13357, 19378, 14054, 27411, 46789, 60843, 88254, 135043, 195886, 142070, 277113, 472999

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 A341312.

It appears that the equation log(a(n)) = 0.265986*n + 1.56445 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(A341313) from linear fit in range 3...10000.

Crossrefs

Cf. A000930, A214551, A341312.

Programs

Maple

RK3:=proc(n) local t1,t2; option remember;
if n <= 2 then 1 else t1:=RK3(n-3)+RK3(n-1);
t2 := min( padic[ordp](RK3(n-3), 2), padic[ordp](RK3(n-1), 2) );
t1/2^t2;
fi;
end;
[seq(RK3(n),n=0..60)];

PARI

a341313(nterms)={my(a=vector(nterms));a[1]=a[2]=1;a[3]=2;for(n=4,nterms,a[n]=(a[n-1]+a[n-3])/2^min(valuation(a[n-1],2),valuation(a[n-3],2)));concat([1],a)};
a341313(60) \\ Hugo Pfoertner, Feb 16 2021