A117584 Generalized Pellian triangle.

1, 1, 2, 1, 3, 5, 1, 4, 7, 12, 1, 5, 9, 17, 29, 1, 6, 11, 22, 41, 70, 1, 7, 13, 27, 53, 99, 169, 1, 8, 15, 32, 65, 128, 239, 408, 1, 9, 17, 37, 77, 157, 309, 577, 985, 1, 10, 19, 42, 89, 186, 379, 746, 1393, 2378

Offset

1,3

Comments

Diagonals of the triangle are composed of the infinite set of Pellian sequences. Right border = A000129. Next diagonal going to the left = A001333 starting (1, 3, 7, 17, ...). A048654 = (1, 4, 9, ...). A048655 = (1, 5, 11, ...). A048693 = (1, 6, 13, ...); and so on.

Links

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Formula

Antidiagonals of the generalized Pellian array. First row of the array = A000129: (1, 2, 5, 12, ...). n-th row of the array starts (1, n+1, ...); as a Pellian sequence.

From G. C. Greubel, Jul 05 2021: (Start)

T(n, k) = P(k) + (n-1)*P(k-1), where P(n) = A000129(n) (square array).

Sum_{k=1..n} T(n-k+1, k) = A117185(n). (End)

Example

First few rows of the triangle are:
  1;
  1, 2;
  1, 3,  5;
  1, 4,  7, 12;
  1, 5,  9, 17, 29;
  1, 6, 11, 22, 41, 70;
  1, 7, 13, 27, 53, 99, 169;
  ...
The triangle rows are antidiagonals of the generalized Pellian array:
  1, 2,  5, 12, 29, ...
  1, 3,  7, 17, 41, ...
  1, 4,  9, 22, 53, ...
  1, 5, 11, 27, 65, ...
  ...
For example, in the row (1, 5, 11, 27, 65, ...), 65 = 2*27 + 11.

Mathematica

T[n_, k_]:= Fibonacci[k, 2] + (n-1)*Fibonacci[k-1, 2];
Table[T[n-k+1, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 05 2021 *)

Prog

(Magma)
P:= func< n | Round( ((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2)) ) >;
T:= func< n, k | P(k) + (n-1)*P(k-1) >;
[T(n-k+1, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 05 2021
(Sage)
def T(n, k): return lucas_number1(k, 2, -1) + (n-1)*lucas_number1(k-1, 2, -1)
flatten([[T(n-k+1, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 05 2021

Crossrefs

Cf. A000129, A001333, A048654, A048655, A048693, A117185.

Sequence in context: A093412 A119355 A076110 * A199847 A047997 A188211

Adjacent sequences: A117581 A117582 A117583 * A117585 A117586 A117587

Keyword

nonn

Author

Gary W. Adamson, Mar 29 2006

Status

approved