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Triangle of differences of Fibonacci numbers.
2

%I #16 Feb 06 2023 17:30:20

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

%T 21,26,29,31,32,33,34,21,34,42,47,50,52,53,54,55,34,55,68,76,81,84,86,

%U 87,88,89,55,89,110,123,131,136,139,141,142,143,144,89,144,178,199,212,220,225,228,230,231,232,233

%N Triangle of differences of Fibonacci numbers.

%C The numbers missing from this triangle form A050939.

%C Row n of this triangle has one more term than row n of A143061.

%C Reversing the rows gives A171730.

%H Michael De Vlieger, <a href="/A171729/b171729.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(n), F(n+1)-F(n-1), ..., F(n+1)-F(2), F(n+1)-F(0).

%e First rows:

%e 1

%e 1 2

%e 1 2 3

%e 2 3 4 5

%e 3 5 6 7 8

%e 5 8 10 11 12 13

%e ...

%p F:= combinat[fibonacci]:

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

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

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

%Y Cf. A000045, A050939, A143061, A171730.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Dec 16 2009