A125608 Triangle read by rows: given the left border = the Lucas numbers, (1, 3, 4, 7, ...), T(n,k) = (n-1,k) + (n-1,k-1).

1, 3, 1, 4, 4, 1, 7, 8, 5, 1, 11, 15, 13, 6, 1, 18, 26, 28, 19, 7, 1, 29, 44, 54, 47, 26, 8, 1, 47, 73, 98, 101, 73, 34, 9, 1, 76, 120, 171, 199, 174, 107, 43, 10, 1, 123, 196, 291, 370, 373, 281, 150, 53, 11, 1, 199, 319, 487, 661, 743, 654, 431, 203, 64, 12, 1, 322, 518, 806, 1148, 1404, 1397, 1085, 634, 267, 76, 13, 1

Offset: 1

Author

Gary W. Adamson, Nov 27 2006

Keywords

• nonn
• tabl

Comments

Row sums = A027973: (1, 4, 9, 21, 46, 99, 209, ...).

Examples

			First few rows of the triangle:
   1;
   3,  1;
   4,  4,  1;
   7,  8,  5,  1;
  11, 15, 13,  6,  1;
  18, 26, 28, 19,  7,  1;
  ...
(6,3) = 28 = 13 + 15 = (5,3) + (5,2).
		

Crossrefs

Cf. A027973.

Programs

Maple

L[1]:=1: L[2]:=3: for n from 3 to 12 do L[n]:=L[n-1]+L[n-2] od: T:=proc(n,k) if k=1 then L[n] elif n=1 then 0 else T(n-1,k)+T(n-1,k-1) fi end: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 01 2007
A000204 := proc(n) if n =1 then RETURN(1) ; elif n = 2 then RETURN(3) ; else RETURN( A000204(n-1)+A000204(n-2)) ; fi ; end ; A125608 := proc(nmax) local a,row,col,anext ; a := [1] ; row := 1 ; while nops(a) < nmax do row := row+1 ; a := [op(a),A000204(row)] ; for col from 2 to row-1 do anext := op(-row,a)+op(-row+1,a) ; a := [op(a),anext] ; od ; a := [op(a),1] ; od ; RETURN(a) ; end ; A125608(80) ; # R. J. Mathar, Jan 07 2007

Mathematica

T[n_, 1] := LucasL[n];
T[n_, k_] /; 2 <= k <= n := T[n, k] = T[n - 1, k] + T[n - 1, k - 1];
T[, ] = 0;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 19 2024 *)

Extensions

More terms from Emeric Deutsch, Jan 01 2007

More terms from R. J. Mathar, Jan 07 2007