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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).

%I A125608 #14 Nov 19 2024 07:27:09
%S A125608 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,
%T A125608 47,73,98,101,73,34,9,1,76,120,171,199,174,107,43,10,1,123,196,291,
%U A125608 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
%N 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).
%C A125608 Row sums = A027973: (1, 4, 9, 21, 46, 99, 209, ...).
%e A125608 First few rows of the triangle:
%e A125608    1;
%e A125608    3,  1;
%e A125608    4,  4,  1;
%e A125608    7,  8,  5,  1;
%e A125608   11, 15, 13,  6,  1;
%e A125608   18, 26, 28, 19,  7,  1;
%e A125608   ...
%e A125608 (6,3) = 28 = 13 + 15 = (5,3) + (5,2).
%p A125608 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
%p A125608 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
%t A125608 T[n_, 1] := LucasL[n];
%t A125608 T[n_, k_] /; 2 <= k <= n := T[n, k] = T[n - 1, k] + T[n - 1, k - 1];
%t A125608 T[_, _] = 0;
%t A125608 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 19 2024 *)
%Y A125608 Cf. A027973.
%K A125608 nonn,tabl
%O A125608 1,2
%A A125608 _Gary W. Adamson_, Nov 27 2006
%E A125608 More terms from _Emeric Deutsch_, Jan 01 2007
%E A125608 More terms from _R. J. Mathar_, Jan 07 2007