A117918 Difference row triangle of the Pell sequence.

1, 1, 2, 2, 3, 5, 2, 4, 7, 12, 4, 6, 10, 17, 29, 4, 8, 14, 24, 41, 70, 8, 12, 20, 34, 58, 99, 169, 8, 16, 28, 48, 82, 140, 239, 408, 16, 24, 40, 68, 116, 198, 338, 577, 985, 16, 32, 56, 96, 164, 280, 478, 816, 1393, 2378, 32, 48, 80, 136, 232, 396, 676, 1154, 1970, 3363, 5741

Offset

1,3

Comments

Leftmost column (1, 1, 2, 2, 4, 4, ...), (A016116); is the inverse binomial transform of the Pell sequence.

References

Raymond Lebois, "Le théorème de Pythagore et ses implications", p. 123, Editions PIM, (1979).

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

Difference rows of the Pell sequence A000129 starting (1, 2, 5, 12, ...) become the diagonals of the triangle.

T(n, n) = A000129(n).

T(n, n-1) = A000129(n) - A000129(n-1).

From G. C. Greubel, Oct 23 2021: (Start)

T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 1) = 2^floor((n-1)/2).

T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n-k, j)*Pell(n-j), where Pell(n) = A000129(n).

Sum_{k=1..n} T(n, k) = Pell(n+1) -2^floor(n/2)*((1 + (-1)^n)/2) - 2^floor((n - 1)/2)*((1 - (-1)^n)/2). (End)

Example

First difference row (1, 3, 7, 17, 41, ...) is the next diagonal.
First few rows of the triangle are:
  1;
  1,  2;
  2,  3,  5;
  2,  4,  7, 12;
  4,  6, 10, 17, 29;
  4,  8, 14, 24, 41, 70;
  8, 12, 20, 34, 58, 99, 169;
  ...

Mathematica

T[n_, k_]:= T[n, k]= If[k==1, 2^Floor[(n-1)/2], T[n, k-1] + T[n-1, k-1]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 22 2021 *)

Prog

(Magma)
Pell:= func< n | Round(((1+Sqrt(2))^n -(1-Sqrt(2))^n)/(2*Sqrt(2))) >;
T:= func< n, k | (&+[ (-1)^j*Binomial(n-k, j)*Pell(n-j): j in [0..n-k]]) >;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 23 2021
(Sage)
def A117918(n, k): return sum( (-1)^j*binomial(n-k, j)*lucas_number1(n-j, 2, -1) for j in (0..n) )
flatten([[A117918(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 23 2021

Crossrefs

Cf. A000129, A016116.

Sequence in context: A175908 A152430 A297495 * A302495 A368255 A368256

Adjacent sequences: A117915 A117916 A117917 * A117919 A117920 A117921

Keyword

nonn,easy

Author

Gary W. Adamson, Apr 02 2006

Status

approved