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

Author

Gary W. Adamson, Apr 02 2006

Keywords

• nonn
• tabl
• easy

Examples

			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;
  ...
		

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

Crossrefs

Column 1 is A016116(n-1).

Diagonals include A000129, A001333, A052542, A002203, A371596.

Programs

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

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 *)

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

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)