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Triangle of differences of Fibonacci numbers.
2

%I #17 Feb 06 2023 17:29:27

%S 1,2,1,3,2,1,5,4,3,2,8,7,6,5,3,13,12,11,10,8,5,21,20,19,18,16,13,8,34,

%T 33,32,31,29,26,21,13,55,54,53,52,50,47,42,34,21,89,88,87,86,84,81,76,

%U 68,55,34,144,143,142,141,139,136,131,123,110,89,55,233,232,231,230,228,225,220,212,199,178,144,89

%N Triangle of differences of Fibonacci numbers.

%C The numbers missing from this triangle form A050939.

%C Reversing the rows gives A171729.

%H Michael De Vlieger, <a href="/A171730/b171730.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)

%F Counting the top row as the first row, the n-th row is

%F F(n+1)-F(0), F(n+1)-F(2), ..., F(n+1)-F(n-1), F(n+1)-F(n).

%e First rows:

%e 1

%e 2 1

%e 3 2 1

%e 5 4 3 2

%e 8 7 6 5 3

%e 13 12 11 10 8 5

%e ...

%p F:= combinat[fibonacci]:

%p T:= (n,k)-> F(n+1)-`if`(k=1, 0, F(k)):

%p seq(seq(T(n,k), k=1..n), n=1..12); # _Alois P. Heinz_, Feb 06 2023

%t Table[Fibonacci[n + 1] - If[k > 1, Fibonacci[k], 0], {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Feb 06 2023 *)

%o (PARI) row(n) = vector(n, k, fibonacci(n+1) - if (k>1, fibonacci(k), 0)); \\ _Michel Marcus_, Feb 06 2023

%Y Cf. A000045, A050939, A143061, A171729.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 16 2009