A193922 Mirror of the triangle A193921.

1, 1, 1, 2, 2, 1, 4, 4, 3, 2, 7, 7, 6, 5, 3, 12, 12, 11, 10, 8, 5, 20, 20, 19, 18, 16, 13, 8, 33, 33, 32, 31, 29, 26, 21, 13, 54, 54, 53, 52, 50, 47, 42, 34, 21, 88, 88, 87, 86, 84, 81, 76, 68, 55, 34, 143, 143, 142, 141, 139, 136, 131, 123, 110, 89, 55, 232, 232, 231

Offset

0,4

Comments

A193922 is obtained by reversing the rows of the triangle A193921.

Also, triangle read by rows: T(n,k) = Fibonacci(n+2) - Fibonacci(k+1) with T(0,0) = 1, 0 <= k <= n. - Arkadiusz Wesolowski, Aug 05 2012

Links

Arkadiusz Wesolowski, Rows n = 0..140 of triangle, flattened

Formula

Write w(n,k) for the triangle at A193921. The triangle at A193922 is then given by w(n,n-k).

G.f.: 1-(x*y-y-x)/((x^2+x-1)*(y^2+y-1)). - Vladimir Kruchinin, Jan 12 2024

Example

First six rows:
   1
   1   1
   2   2   1
   4   4   3   2
   7   7   6   5   3
  12  12  11  10   8   5

Mathematica

z = 12;
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;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193921 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* this sequence *)
Factor[w[7, x]]
Factor[w[8, x]]
Table[Expand[p[n, x]], {n, 0, 4}]
Table[Expand[q[n, x]], {n, 0, 4}]
Prepend[Flatten@Table[Fibonacci[n + 2] - Fibonacci[k + 1], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Aug 05 2012 *)

Crossrefs

Cf. A193921.

Sequence in context: A182222 A225639 A110664 * A319534 A061436 A214095

Adjacent sequences: A193919 A193920 A193921 * A193923 A193924 A193925

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 09 2011

Status

approved