A193999 Mirror of the triangle A094585.

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, 34, 21, 87, 86, 84, 81, 76, 68, 55, 34, 142, 141, 139, 136, 131, 123, 110, 89, 55, 231, 230, 228, 225, 220, 212, 199, 178, 144, 89, 375, 374, 372, 369, 364, 356, 343

Offset: 1

Author

Clark Kimberling, Aug 11 2011

Keywords

• nonn

Comments

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

Examples

			First six rows:
   1;
   3,  2;
   6,  5,  3;
  11, 10,  8,  5;
  19, 18, 16, 13,  8;
  32, 31, 29, 26, 21, 13;
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..11325

Crossrefs

Cf. A094585.

Programs

GAP

Flat(List([1..11],n->Reversed(List([1..n],k->Fibonacci(n+3)-Fibonacci(n-k+3))))); # Muniru A Asiru, Apr 28 2019

Mathematica

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := x*q[n - 1, x] + 1; q[0, n_] := 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A094585 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193999 *)
(* alternate program *)
Table[Fibonacci[n+3]-Fibonacci[k+2], {n,1,10}, {k,1,n}] //TableForm (* Rigoberto Florez, Oct 03 2019 *)

Formula

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

T(n,k) = Fibonacci(n+3) - Fibonacci(k+2) for n > 0 and 1 <= k <= n. - Rigoberto Florez, Oct 03 2019

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

Extensions

Offset 1 from Muniru A Asiru, Apr 29 2019