A164981 A triangle with Pell numbers in the first column.

1, 2, 1, 5, 3, 1, 12, 10, 4, 1, 29, 30, 16, 5, 1, 70, 87, 56, 23, 6, 1, 169, 245, 185, 91, 31, 7, 1, 408, 676, 584, 334, 136, 40, 8, 1, 985, 1836, 1784, 1158, 546, 192, 50, 9, 1, 2378, 4925, 5312, 3850, 2052, 834, 260, 61, 10, 1, 5741, 13079, 15497, 12386, 7342, 3366, 1212, 341, 73, 11, 1

Offset: 1

Author

Mark Dols, Sep 03 2009

Keywords

• nonn
• tabl

Comments

Rows sum up to A000244 (powers of 3), diagonals to A001654 (golden rectangles).

Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A210557, i.e. the numbers are the same just read row-wise in the opposite direction. [Christine Bessenrodt, Jul 20 2012]

Subtriangle of (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2013

Examples

			Triangle begins
  1
  2,1
  5,3,1
  12,10,4,1
  29,30,16,5,1
  70,87,56,23,6,1
  169,245,185,91,31,7,1
  ...
From _Philippe Deléham_, Oct 10 2013: (Start)
Triangle (0, 2, 1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...):
  1
  0, 1
  0, 2, 1
  0, 5, 3, 1
  0, 12, 10, 4, 1
  0, 29, 30, 16, 5, 1
  0, 70, 87, 56, 23, 6, 1
  0, 169, 245, 185, 91, 31, 7, 1
  ... (End)
		

Crossrefs

Cf. A000244, A001654, A164975, A164976, A210557.

Programs

Maple

A164981 := proc(n,k) option remember; if n <1 or k<1 or k>n then 0; elif n = 1 then 1; else 2*procname(n-1,k)+procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc:

Mathematica

T[n_, k_] := T[n, k] = Which[n < 1 || k < 1 || k > n, 0, n == 1, 1, True, 2*T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1]];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 06 2023 *)

PARI

T(n,k) = if ((n==1) && (k==1), return(1)); if ((n<=0) || (k<=0) || (nMichel Marcus, Feb 01 2023

Formula

From R. J. Mathar, Jan 27 2011: (Start)

T(1,1) =1. T(n,k)=0 if n<1 or k<1 or k>n. T(n,k) = 2*T(n-1,k)+T(n-1,k-1)+T(n-2,k)-T(n-2,k-1) otherwise.

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

T(n,n-1) = n.

T(n,n-2) = A052905(n-2).

T(n,2) = A026937(n-2). (End)

G.f. x*y/(1-2*x-x^2+x^2*y-x*y). - R. J. Mathar, Aug 11 2015

Extensions

Rows 10-11 from Michel Marcus, Feb 01 2023