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A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

%I A173284 #13 Aug 15 2025 02:27:06
%S A173284 1,1,2,1,3,1,5,2,1,8,3,1,13,5,2,21,8,3,1,34,13,5,2,1,55,21,8,3,1,89,
%T A173284 34,13,5,2,1,144,55,21,8,3,1,233,89,34,13,5,2,1,377,144,55,21,8,3,1,
%U A173284 610,233,89,34,13,5,2,1
%N A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
%C A173284 The row sums equal A052952.
%C A173284 Let the triangle = M. Then lim_{n->infinity} M^n = A173285 as a left-shifted vector.
%C A173284 A173284 * [1, 2, 3, ...] = A054451: (1, 1, 4, 5, 12, 17, 33, ...). - _Gary W. Adamson_, Mar 03 2010
%C A173284 From _Johannes W. Meijer_, Sep 05 2013: (Start)
%C A173284 Triangle read by rows formed from antidiagonals of triangle A104762.
%C A173284 The diagonal sums lead to A004695. (End)
%F A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
%F A173284 From _Johannes W. Meijer_, Sep 05 2013: (Start)
%F A173284 T(n,k) = A000045(n-2*k+1), n >= 0 and 0 <= k <= floor(n/2).
%F A173284 T(n,k) = A104762(n-k, k). (End)
%e A173284 First few rows of the triangle:
%e A173284     1;
%e A173284     1;
%e A173284     2,   1;
%e A173284     3,   1;
%e A173284     5,   2,  1;
%e A173284     8,   3,  1;
%e A173284    13,   5,  2,  1;
%e A173284    21,   8,  3,  1;
%e A173284    34,  13,  5,  2,  1;
%e A173284    55,  21,  8,  3,  1;
%e A173284    89,  34, 13,  5,  2, 1;
%e A173284   144,  55, 21,  8,  3, 1;
%e A173284   233,  89, 34, 13,  5, 2, 1;
%e A173284   377, 144, 55, 21,  8, 3, 1;
%e A173284   610, 233, 89, 34, 13, 5, 2, 1;
%e A173284   ...
%p A173284 T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # _Johannes W. Meijer_, Sep 05 2013
%Y A173284 Cf. A000045, A004695, A052952, A054451, A173285.
%Y A173284 Cf. (Similar triangles) A008315 (Catalan), A011973 (Pascal), A102541 (Losanitsch), A122196 (Fractal), A122197 (Fractal), A128099 (Pell-Jacobsthal), A152198, A152204, A207538, A209634.
%K A173284 nonn,tabf
%O A173284 0,3
%A A173284 _Gary W. Adamson_, Feb 14 2010
%E A173284 Term a(15) corrected by _Johannes W. Meijer_, Sep 05 2013