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

0,5

Comments

Deleting the right border gives triangle A117895.

Links

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

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)

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A000129, A078343, A117584, A117895.

Sequence in context: A368153 A355668 A023990 * A177762 A109380 A167754

Adjacent sequences: A117891 A117892 A117893 * A117895 A117896 A117897

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 30 2006

Status

approved