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A204922 Ordered differences of Fibonacci numbers.

%I A204922 #43 Feb 19 2025 17:31:49
%S A204922 1,2,1,4,3,2,7,6,5,3,12,11,10,8,5,20,19,18,16,13,8,33,32,31,29,26,21,
%T A204922 13,54,53,52,50,47,42,34,21,88,87,86,84,81,76,68,55,34,143,142,141,
%U A204922 139,136,131,123,110,89,55,232,231,230,228,225,220,212,199,178
%N A204922 Ordered differences of Fibonacci numbers.
%C A204922 For a guide to related sequences, see A204892. For numbers not in A204922, see A050939.
%C A204922 From _Emanuele Munarini_, Mar 29 2012: (Start)
%C A204922 Diagonal elements = Fibonacci numbers F(n+1) (A000045)
%C A204922 First column = Fibonacci numbers - 1 (A000071);
%C A204922 Second column = Fibonacci numbers - 2 (A001911);
%C A204922 Row sums = n*F(n+3) - F(n+2) + 2 (A014286);
%C A204922 Central coefficients = F(2*n+1) - F(n+1) (A096140).
%C A204922 (End)
%H A204922 G. C. Greubel, <a href="/A204922/b204922.txt">Rows n=1..100 of triangle, flattened</a>
%F A204922 From _Emanuele Munarini_, Mar 29 2012: (Start)
%F A204922 T(n,k) = Fibonacci(n+2) - Fibonacci(k+1).
%F A204922 T(n,k) = Sum_{i=k..n} Fibonacci(i+1). (End)
%e A204922 a(1) = s(2) - s(1) = F(3) - F(2) = 2-1 = 1, where F=A000045;
%e A204922 a(2) = s(3) - s(1) = F(4) - F(2) = 3-1 = 2;
%e A204922 a(3) = s(3) - s(2) = F(4) - F(3) = 3-2 = 1;
%e A204922 a(4) = s(4) - s(1) = F(5) - F(2) = 5-1 = 4.
%e A204922 From _Emanuele Munarini_, Mar 29 2012: (Start)
%e A204922 Triangle begins:
%e A204922    1;
%e A204922    2,  1;
%e A204922    4,  3,  2;
%e A204922    7,  6,  5,  3;
%e A204922   12, 11, 10,  8,  5;
%e A204922   20, 19, 18, 16, 13,  8;
%e A204922   33, 32, 31, 29, 26, 21, 13;
%e A204922   54, 53, 52, 50, 47, 42, 34, 21;
%e A204922   88, 87, 86, 84, 81, 76, 68, 55, 34;
%e A204922   ... (End)
%t A204922 (See the program at A204924.)
%o A204922 (Maxima) create_list(fib(n+3)-fib(k+2),n,0,20,k,0,n); /* _Emanuele Munarini_, Mar 29 2012 */
%o A204922 (Magma) /* As triangle */ [[Fibonacci(n+2)-Fibonacci(k+1) : k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Aug 04 2015
%o A204922 (PARI) {T(n,k) = fibonacci(n+2) - fibonacci(k+1)};
%o A204922 for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 03 2019
%o A204922 (Sage) [[fibonacci(n+2) - fibonacci(k+1) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Feb 03 2019
%Y A204922 Cf. A204924, A204892.
%K A204922 nonn,tabl
%O A204922 1,2
%A A204922 _Clark Kimberling_, Jan 21 2012