A106585 Triangle read by rows: even-numbered rows of A106580.

1, 1, 2, 2, 1, 2, 5, 7, 7, 1, 2, 5, 13, 22, 29, 29, 1, 2, 5, 13, 34, 65, 101, 130, 130, 1, 2, 5, 13, 34, 89, 185, 322, 481, 611, 611, 1, 2, 5, 13, 34, 89, 233, 514, 973, 1613, 2354, 2965, 2965, 1, 2, 5, 13, 34, 89, 233, 610, 1405, 2837, 5090, 8185, 11761, 14726, 14726

Offset: 0

Author

N. J. A. Sloane, May 30 2005

Keywords

• nonn
• tabf
• easy

Examples

			Irregular triangle begins as:
  1;
  1, 2, 2;
  1, 2, 5,  7,  7;
  1, 2, 5, 13, 22, 29,  29;
  1, 2, 5, 13, 34, 65, 101, 130,  130;
  1, 2, 5, 13, 34, 89, 185, 322,  481,  611,  611;
  1, 2, 5, 13, 34, 89, 233, 514,  973, 1613, 2354, 2965, 2965;
  1, 2, 5, 13, 34, 89, 233, 610, 1405, 2837, 5090, 8185, 11761, 14726, 14726;
		

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Crossrefs

Cf. A106580, A106595.

Programs

Maple

A106580:= proc(n,k) option remember; if k =0 then 1; else A106580(n,k-1) + add(A106580(n-2*i, k-i), i=1..min(k,floor(n/2),n-k)); fi; end: for n from 0 to 18 by 2 do for k from 0 to n do printf("%d, ",A106580(n,k)); od; od; # R. J. Mathar, Aug 10 2007

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, T[n, k-1] + Sum[T[n-2*j, k-j], {j, 1, Min[k, Floor[n/2], n-k]}]];  (* T(n, k) = A106580; T(2*n, k) = A106585 *)
Table[T[2*n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Sep 07 2021 *)

Sage

@CachedFunction
def T(n, k): # T(n, k) = A106580; T(2*n, k) = A106585
    if (k<0): return 0
    elif (k==0): return 1
    else: return T(n, k-1) + sum( T(n-2*j, k-j) for j in (1..min(k, n//2, n-k)))
flatten([[T(2*n, k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Sep 07 2021

Formula

T(n, k) = A106580(2*n, k).

Extensions

More terms from R. J. Mathar, Aug 10 2007