A125172 Triangle T(n,k) with partial column sums of the even-indexed Fibonacci numbers.

1, 3, 1, 8, 4, 1, 21, 12, 5, 1, 55, 33, 17, 6, 1, 144, 88, 50, 23, 7, 1, 377, 232, 138, 73, 30, 8, 1, 987, 609, 370, 211, 103, 38, 9, 1, 2584, 1596, 979, 581, 314, 141, 47, 10, 1

Offset

1,2

Comments

"Partial column sums" means the 1st column consists of the even-indexed Fibonacci numbers, the 2nd column shows the partial sums of the first column, the 3rd column the partial sums of the 2nd, etc. - R. J. Mathar, Sep 06 2011

Mirror of the fission triangle A193667, as in the Mathematica program below. - Clark Kimberling, Aug 11 2011

Links

Table of n, a(n) for n=1..45.

Formula

T(n,1) = A001906(n).

T(n,k) = T(n-1,k-1) + T(n-1,k), k > 1.

From R. J. Mathar, Sep 06 2011: (Start)

T(n,k) = A125171(n,k), i.e., A125171 without column k=0.

Conjecture: T(n,k) = T(n,k-1) - A121460(n+1,k). (End)

Example

First few rows of the triangle:
    1;
    3,  1;
    8,  4,  1;
   21, 12,  5,  1;
   55, 33, 17,  6,  1;
  144, 88, 50, 23,  7,  1;
  ...

Mathematica

z = 11;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
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}]]  (* A193667 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* this sequence *)
(* Clark Kimberling, Aug 11 2011 *)

Crossrefs

Cf. A105693 (row sums), A125171, A193667.

Sequence in context: A054506 A101026 A055249 * A344381 A073732 A021318

Adjacent sequences: A125169 A125170 A125171 * A125173 A125174 A125175

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Nov 22 2006

Status

approved