|
%I #24 Oct 17 2019 14:27:20
%S 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 52,53,34,55,68,76,81,84,86,87,55,89,110,123,131,136,139,141,142,89,
%U 144,178,199,212,220,225,228,230,231,144,233,288,322,343,356,364,369,372
%N Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers.
%C 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 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 Muniru A Asiru, <a href="/A094585/b094585.txt">Rows n=1..150 of triangle, flattened</a>
%F 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.
%e Rows 1 to 5:
%e 1;
%e 2, 3;
%e 3, 5, 6;
%e 5, 8, 10, 11;
%e 8, 13, 16, 18, 19;
%e T(5,4) = F(8) - F(4) = 21 - 3 = 18;
%e T(5,4) = F(6) + F(5) + F(4) + F(3) = 8 + 5 + 3 + 2 = 18.
%t See A193999.
%t Table[Fibonacci[n+3]-Fibonacci[n+3-k],{n,1,20}, {k,1,n}]//TableForm (* _Rigoberto Florez_, Oct 03 2019 *)
%o (GAP) Flat(List([1..11],n->List([1..n],k->Fibonacci(n+3)-Fibonacci(n-k+3)))); # _Muniru A Asiru_, Apr 28 2019
%Y Cf. A000045, A094584, A094586, A193999.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, May 13 2004
|
