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

0,3

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.

Links

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

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)

Example

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).

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 *)

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-1], n in [0..12]]; // 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

Crossrefs

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

Sequence in context: A211782 A211701 A183110 * A188002 A186974 A286312

Adjacent sequences: A117892 A117893 A117894 * A117896 A117897 A117898

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 30 2006

Extensions

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

Status

approved