A027037 Diagonal sum of left-justified array T given by A027023.

1, 1, 2, 3, 3, 6, 7, 11, 16, 21, 33, 48, 65, 101, 146, 203, 311, 450, 635, 963, 1396, 1989, 2993, 4348, 6233, 9329, 13574, 19543, 29135, 42446, 61303, 91123, 132884, 192377, 285309, 416384, 603925, 894069, 1305618, 1896495, 2803611, 4096182, 5957183, 8796287

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Programs

Maple

T:= proc(n, k) option remember;
      if n<0 or k>2*n then 0
    elif k<3 or k=2*n then 1
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n-k,k), k=0..n), n=0..30); # G. C. Greubel, Nov 05 2019

Mathematica

T[n_, k_]:= T[n, k]= If[n<0 || k>2*n, 0, If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n-k, k], {k, 0, n}], {n,0,30}] (* G. C. Greubel, Nov 05 2019 *)

Sage

@CachedFunction
def T(n, k):
    if (n<0 or k>2*n): return 0
    elif (k<3 or k==2*n): return 1
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n-k, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 05 2019

Formula

a(n) = Sum_{k=0..n} A027023(n-k, k). - Sean A. Irvine, Oct 22 2019

Extensions

More terms from Sean A. Irvine, Oct 21 2019