A117895 Triangle T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.

1, 1, 2, 1, 3, 3, 1, 4, 4, 8, 1, 5, 5, 11, 19, 1, 6, 6, 14, 26, 46, 1, 7, 7, 17, 33, 63, 111, 1, 8, 8, 20, 40, 80, 152, 268, 1, 9, 9, 23, 47, 97, 193, 367, 647, 1, 10, 10, 26, 54, 114, 234, 466, 886, 1562, 1, 11, 11, 29, 61, 131, 275, 565, 1125, 2139, 3771, 1, 12, 12, 32, 68, 148, 316, 664, 1364, 2716, 5164, 9104

Offset: 0

Author

Gary W. Adamson, Mar 30 2006

Keywords

• nonn
• tabl

Comments

Successive deletions of the right borders of triangle A117894 produces triangles whose row sums = generalized Pell sequences starting (1, 2...), (1, 3...), (1, 4...); etc. Row sums of A117894 = A000129: (1, 2, 5...). Row sums of A117895 = A001333: (1, 3, 7...). Deletion of the border of A117895 would produce a triangle with row sums of the Pell sequence A048654 (1, 4, 9...); and so on.

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  1, 3, 3;
  1, 4, 4,  8;
  1, 5, 5, 11, 19;
  1, 6, 6, 14, 26, 46;
  1, 7, 7, 17, 33, 63, 111;
  1, 8, 8, 20, 40, 80, 152, 268;
...
Row 4, (1, 4, 4, 8) is produced by adding (0, 1, 1, 3) to row 4 of A117894: (1, 3, 3, 5).
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000129, A001333, A048654, A048655, A048693, A117894.

Programs

Magma

Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
[k eq 0 select 1 else (k-n)*Pell(k+1) + (3*n-3*k+1)*Pell(k): k in [0..n-1], n in [0..12]]; // G. C. Greubel, Sep 27 2021

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, (k-n)*Fibonacci[k+1, 2] + (3*n-3*k +1)*Fibonacci[k, 2]]; Table[T[n, k], {n,0,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Sep 27 2021 *)

Sage

def P(n): return lucas_number1(n, 2, -1)
def A117895(n,k): return 1 if (k==0) else (k-n)*P(k+1) + (3*n-3*k+1)*P(k)
flatten([[A117895(n,k) for k in (0..n-1)] for n in (0..12)]) # G. C. Greubel, Sep 27 2021

Formula

Delete right border of triangle A117894. Alternatively, let row 1 = 1 and using the heading 0, 1, 1, 3, 7, 17, 41, 99, 239...(i.e. A001333 starting with 0, 1, 1, 3...); add the first n terms of the heading to n-th row of triangle A117894.

From G. C. Greubel, Sep 27 2021: (Start)

T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1.

T(n, 1) = n+1 for n >= 1.

T(n, 2) = n+1 for n >= 2.

T(n, n) = 2*[n=0] + A078343(n). (End)

Extensions

New name and more terms added by G. C. Greubel, Sep 27 2021