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).
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
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Apr 02 2006
STATUS
approved