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

1,2

Comments

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

Links

Table of n, a(n) for n=1..78.

Example

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).

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 *)

Crossrefs

Cf. A027973.

Sequence in context: A246354 A286625 A129246 * A182001 A099813 A376782

Adjacent sequences: A125605 A125606 A125607 * A125609 A125610 A125611

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 27 2006

Extensions

More terms from Emeric Deutsch, Jan 01 2007

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

Status

approved