A117894 Triangle, row sums = Pell numbers, A000129.

1, 1, 1, 1, 2, 2, 1, 3, 3, 5, 1, 4, 4, 8, 12, 1, 5, 5, 11, 19, 29, 1, 6, 6, 14, 26, 46, 70, 1, 7, 7, 17, 33, 63, 111, 169, 1, 8, 8, 20, 40, 80, 152, 268, 408, 1, 9, 9, 23, 47, 97, 193, 367, 647, 985

Offset: 0

Author

Gary W. Adamson, Mar 30 2006

Keywords

• nonn
• tabl

Comments

Deleting the right border gives triangle A117895.

Examples

			First few rows of the triangle are:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 3,  5;
  1, 4, 4,  8, 12;
  1, 5, 5, 11, 19, 29;
  1, 6, 6, 14, 26, 46,  70;
  1, 7, 7, 17, 33, 63, 111, 169;
...
Row 4 of A117584 = (1, 4, 7, 12). Difference terms (1, 3, 3, 5) = row 4 of A117894.
		

Links

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

Crossrefs

Cf. A000129, A078343, A117584, A117895.

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], 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}]//Flatten (* G. C. Greubel, Sep 27 2021 *)

Sage

def P(n): return lucas_number1(n, 2, -1)
def A117894(n,k): return 1 if (k==0) else (k-n)*P(k+1) + (3*n-3*k+1)*P(k)
flatten([[A117894(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 27 2021

Formula

Rows are composed of difference terms of triangle A117584.

Rows sum to Pell numbers, A000129.

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, k) = A117584(n+1, k+1) - A117584(n+1, k) with T(n, 0) = 1.

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

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

T(n, n) = [n=0] + A000129(n).

T(n, n-1) = 2*[n=0] + A078343(n). (End)