A193999 - OEIS

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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

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OFFSET

1,2

COMMENTS

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

LINKS

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

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

EXAMPLE

First six rows:

3, 2;

6, 5, 3;

11, 10, 8, 5;

19, 18, 16, 13, 8;

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

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 *)

PROG

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

CROSSREFS

Cf. A094585.

Sequence in context: A297878 A234922 A049777 * A210971 A212000 A058401

Adjacent sequences: A193996 A193997 A193998 * A194000 A194001 A194002

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 11 2011

EXTENSIONS

Offset 1 from Muniru A Asiru, Apr 29 2019

STATUS

approved