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A193922 Mirror of the triangle A193921. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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).

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}]]  (* A193922 *)

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

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Last modified June 6 04:15 EDT 2020. Contains 334859 sequences. (Running on oeis4.)