A026742 a(n) = T(n, floor(n/2)), T given by A026736.

1, 1, 2, 3, 6, 11, 21, 43, 79, 173, 309, 707, 1237, 2917, 5026, 12111, 20626, 50503, 85242, 211263, 354080, 885831, 1476368, 3720995, 6173634, 15652239, 25873744, 65913927, 108628550, 277822147, 456710589, 1171853635, 1922354351

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A026736.

Programs

GAP

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
Flat(List([0..20], n-> T(n,Int(n/2)) )); # G. C. Greubel, Jul 19 2019

Mathematica

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/2, T[n-1, k-1] +T[n-2, k-1] +T[n-1, k], T[n-1, k-1] + T[n-1, k]]]; Table[T[n, Floor[n/2]], {n,0,40}] (* G. C. Greubel, Jul 19 2019 *)

PARI

T(n,k) = if(k==n || k==0, 1, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(20, n, n--; T(n, n\2)) \\ G. C. Greubel, Jul 19 2019

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[T(n, floor(n/2)) for n in (0..40)] # G. C. Greubel, Jul 19 2019

Formula

a(n) ~ phi^(3*n/2 - (7 + (-1)^n)/4) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019