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A094585 Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers.

%I A094585 #32 Aug 08 2025 03:20:20
%S A094585 1,2,3,3,5,6,5,8,10,11,8,13,16,18,19,13,21,26,29,31,32,21,34,42,47,50,
%T A094585 52,53,34,55,68,76,81,84,86,87,55,89,110,123,131,136,139,141,142,89,
%U A094585 144,178,199,212,220,225,228,230,231,144,233,288,322,343,356,364,369,372,374,375
%N A094585 Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers.
%C A094585 Row sums = (1,5,14,34,74,...) = A094584. Alternating row sums = (1,1,4,4,12,12,...) given by F(m+1)-1 if m is even and F(m+2)-1 if m is odd. Central numbers = (1,5,16,47,...) = A094586.
%C A094585 Let p(n,x) = Sum_{k=0..n} F(k+1)*x^(n-k) and q(n,x) = x * q(n-1,x)+1, with q(0,x)=1. Then A094585 is the fission of sequence (p(n,x)) by sequence (q(n,x)); see A193842 for the definition of fission. A094585 is the mirror of A193999. - _Clark Kimberling_, Aug 11 2011
%H A094585 Muniru A Asiru, <a href="/A094585/b094585.txt">Rows n=1..150 of triangle, flattened</a>
%F A094585 T(n, k) = F(n+3) - F(n+3-k) = F(n+1) + F(n) + ... + F(n+2-k), for k=1..n; n >= 1.
%F A094585 G.f.: x*y*(x*y+x+1)/((1-y*x)*(x^2+x-1)*(x^2*y^2+x*y-1)). - _Vladimir Kruchinin_, Jun 20 2025
%e A094585 Rows 1 to 5:
%e A094585   1;
%e A094585   2,  3;
%e A094585   3,  5,  6;
%e A094585   5,  8, 10, 11;
%e A094585   8, 13, 16, 18, 19;
%e A094585 T(5,4) = F(8) - F(4) = 21 - 3 = 18;
%e A094585 T(5,4) = F(6) + F(5) + F(4) + F(3) = 8 + 5 + 3 + 2 = 18.
%t A094585 (* See A193999. *)
%t A094585 Table[Fibonacci[n+3]-Fibonacci[n+3-k],{n,1,20}, {k,1,n}]//TableForm (* _Rigoberto Florez_, Oct 03 2019 *)
%o A094585 (GAP) Flat(List([1..11],n->List([1..n],k->Fibonacci(n+3)-Fibonacci(n-k+3)))); # _Muniru A Asiru_, Apr 28 2019
%Y A094585 Cf. A000045, A094584, A094586, A193999.
%K A094585 nonn,tabl
%O A094585 1,2
%A A094585 _Clark Kimberling_, May 13 2004