%I #28 Oct 11 2019 06:32:02

%S 1,3,2,6,5,3,11,10,8,5,19,18,16,13,8,32,31,29,26,21,13,53,52,50,47,42,

%T 34,21,87,86,84,81,76,68,55,34,142,141,139,136,131,123,110,89,55,231,

%U 230,228,225,220,212,199,178,144,89,375,374,372,369,364,356,343

%N Mirror of the triangle A094585.

%C A193999 is obtained by reversing the rows of the triangle A094585.

%H Muniru A Asiru, <a href="/A193999/b193999.txt">Table of n, a(n) for n = 1..11325</a>

%F Write w(n,k) for the triangle at A094585. The triangle at A094585 is then given by w(n,n-k).

%F T(n,k) = Fibonacci(n+3) - Fibonacci(k+2) for n > 0 and 1 <= k <= n. - _Rigoberto Florez_, Oct 03 2019

%e First six rows:

%e 1;

%e 3, 2;

%e 6, 5, 3;

%e 11, 10, 8, 5;

%e 19, 18, 16, 13, 8;

%e 32, 31, 29, 26, 21, 13;

%t z = 11;

%t p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

%t q[n_, x_] := x*q[n - 1, x] + 1; q[0, n_] := 1;

%t p1[n_, k_] := Coefficient[p[n, x], x^k];

%t p1[n_, 0] := p[n, x] /. x -> 0;

%t d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]

%t h[n_] := CoefficientList[d[n, x], {x}]

%t TableForm[Table[Reverse[h[n]], {n, 0, z}]]

%t Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A094585 *)

%t TableForm[Table[h[n], {n, 0, z}]]

%t Flatten[Table[h[n], {n, -1, z}]] (* A193999 *)

%t (* alternate program *)

%t Table[Fibonacci[n+3]-Fibonacci[k+2], {n,1,10}, {k,1,n}] //TableForm (* _Rigoberto Florez_, Oct 03 2019 *)

%o (GAP) Flat(List([1..11],n->Reversed(List([1..n],k->Fibonacci(n+3)-Fibonacci(n-k+3))))); # _Muniru A Asiru_, Apr 28 2019

%Y Cf. A094585.

%K nonn

%O 1,2

%A _Clark Kimberling_, Aug 11 2011

%E Offset 1 from _Muniru A Asiru_, Apr 29 2019