A106580 -id:A106580 - cp's OEIS frontend

A106595 Triangle read by rows: odd-numbered rows of A106580.

1, 1, 1, 2, 3, 3, 1, 2, 5, 9, 12, 12, 1, 2, 5, 13, 26, 41, 53, 53, 1, 2, 5, 13, 34, 73, 129, 194, 247, 247, 1, 2, 5, 13, 34, 89, 201, 386, 645, 945, 1192, 1192, 1, 2, 5, 13, 34, 89, 233, 546, 1117, 2021, 3266, 4705, 5897, 5897, 1, 2, 5, 13, 34, 89, 233, 610, 1469, 3157, 6082, 10593, 16737, 23826, 29723, 29723

Offset: 0

N. J. A. Sloane, May 30 2005

			Irregular triangle begins as:
  1, 1;
  1, 2, 3,  3;
  1, 2, 5,  9, 12, 12;
  1, 2, 5, 13, 26, 41,  53,  53;
  1, 2, 5, 13, 34, 73, 129, 194,  247,  247;
  1, 2, 5, 13, 34, 89, 201, 386,  645,  945, 1192, 1192;
  1, 2, 5, 13, 34, 89, 233, 546, 1117, 2021, 3266, 4705, 5897, 5897;

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

Cf. A106580, A106585.

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 1 to 13 by 2 do for k from 0 to n do printf("%d, ",A106580(n,k)) ; od ; od ; # R. J. Mathar, May 02 2007

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+1, k) = A106595 *)
Table[T[2*n+1, k], {n, 0, 12}, {k, 0, 2*n+1}]//Flatten (* G. C. Greubel, Sep 08 2021 *)

@CachedFunction
def T(n, k): # T(n, k) = A106580; T(2*n+1, k) = A106595
    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+1, k) for k in (0..2*n+1)] for n in (0..12)]) # G. C. Greubel, Sep 08 2021

T(n, k) = A106580(2*n+1, k).

More terms from R. J. Mathar, May 02 2007

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

Original entry on oeis.org

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

N. J. A. Sloane, May 30 2005

			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;

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

Cf. A106580, A106595.

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

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 *)

@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

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

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

cp's OEIS Frontend

A106595 Triangle read by rows: odd-numbered rows of A106580.

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