A078446 a(1)=a(2)=1; a(n)=a(n-2)/2 if a(n-2) is even, a(n)=a(n-1)+a(n-2) otherwise.

1, 1, 2, 3, 1, 4, 5, 2, 7, 1, 8, 9, 4, 13, 2, 15, 1, 16, 17, 8, 25, 4, 29, 2, 31, 1, 32, 33, 16, 49, 8, 57, 4, 61, 2, 63, 1, 64, 65, 32, 97, 16, 113, 8, 121, 4, 125, 2, 127, 1, 128, 129, 64, 193, 32, 225, 16, 241, 8, 249, 4, 253, 2, 255, 1, 256, 257, 128, 385, 64, 449, 32, 481, 16, 497

Offset: 1

Author

Benoit Cloitre, Dec 31 2002

Keywords

• nonn

Comments

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Programs

Maple

a:= proc(n) option remember;
      if n < 3 then 1
    elif `mod`(procname(n-2), 2) = 0 then procname(n-2)/2
    else procname(n-1) + procname(n-2)
      fi
    end:
seq(a(n), n=1..80); # G. C. Greubel, Nov 07 2019

Mathematica

a[n_]:= a[n]= If[n<3, 1, If[EvenQ[a[n-2]], a[n-2]/2, a[n-1]+a[n-2]]];
Table[a[n], {n, 80}] (* G. C. Greubel, Nov 07 2019 *)
nxt[{a_,b_}]:={b,If[EvenQ[a],a/2,a+b]}; NestList[nxt,{1,1},80][[;;,1]] (* Harvey P. Dale, Jul 18 2025 *)

PARI

a(n) = if(n<3, 1, if(a(n-2)%2==0, a(n-2)/2, a(n-1) + a(n-2) )); \\ G. C. Greubel, Nov 07 2019

Sage

@CachedFunction
def a(n):
    if (n<3): return 1
    elif (a(n-2)%2==0): return a(n-2)/2
    else: return a(n-1) + a(n-2)
[a(n) for n in (1..80)] # G. C. Greubel, Nov 07 2019

Formula

a(n^2)=2^n-1; a(n^2+1)=1; a(n^2+2)=2^n; a(n^2+3)=2^n+1; a(n^2+4)=2^(n-1); a(n^2+5)=3*2^n+1 ...; inequality : a(n)/2^sqrt(n) <2

Sum(k=1, n^2, a(k)) = 2*(n-2)*2^n + n*(n+1)/2 + 4