cp's OEIS Frontend

A193999 Mirror of the triangle A094585.

%I A193999 #32 Jun 20 2025 08:37:01
%S A193999 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 A193999 34,21,87,86,84,81,76,68,55,34,142,141,139,136,131,123,110,89,55,231,
%U A193999 230,228,225,220,212,199,178,144,89,375,374,372,369,364,356,343
%N A193999 Mirror of the triangle A094585.
%C A193999 A193999 is obtained by reversing the rows of the triangle A094585.
%H A193999 Muniru A Asiru, <a href="/A193999/b193999.txt">Table of n, a(n) for n = 1..11325</a>
%F A193999 Write w(n,k) for the triangle at A094585.  The triangle at A094585 is then given by w(n,n-k).
%F A193999 T(n,k) = Fibonacci(n+3) - Fibonacci(k+2) for n > 0 and 1 <= k <= n. - _Rigoberto Florez_, Oct 03 2019
%F A193999 G.f.: x*y*(x*y+x+1)/((1-x)*(x^2+x-1)*(x^2*y^2+x*y-1)). - _Vladimir Kruchinin_, Jun 20 2025
%e A193999 First six rows:
%e A193999    1;
%e A193999    3,  2;
%e A193999    6,  5,  3;
%e A193999   11, 10,  8,  5;
%e A193999   19, 18, 16, 13,  8;
%e A193999   32, 31, 29, 26, 21, 13;
%t A193999 z = 11;
%t A193999 p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A193999 q[n_, x_] := x*q[n - 1, x] + 1; q[0, n_] := 1;
%t A193999 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193999 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193999 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193999 h[n_] := CoefficientList[d[n, x], {x}]
%t A193999 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193999 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A094585 *)
%t A193999 TableForm[Table[h[n], {n, 0, z}]]
%t A193999 Flatten[Table[h[n], {n, -1, z}]]  (* A193999 *)
%t A193999 (* alternate program *)
%t A193999 Table[Fibonacci[n+3]-Fibonacci[k+2], {n,1,10}, {k,1,n}] //TableForm (* _Rigoberto Florez_, Oct 03 2019 *)
%o A193999 (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 A193999 Cf. A094585.
%K A193999 nonn
%O A193999 1,2
%A A193999 _Clark Kimberling_, Aug 11 2011
%E A193999 Offset 1 from _Muniru A Asiru_, Apr 29 2019