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

1,2

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

Links

Table of n, a(n) for n=1..66.

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

Example

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)

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 *)

Prog

(PARI) T(n, k) = if ((n==1) && (k==1), return(1)); if ((n<=0) || (k<=0) || (n<k), return(0)); 2*T(n-1, k)+T(n-1, k-1)+T(n-2, k)-T(n-2, k-1); \\ Michel Marcus, Feb 01 2023

Crossrefs

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

Sequence in context: A130197 A106513 A054446 * A366858 A047858 A125171

Adjacent sequences: A164978 A164979 A164980 * A164982 A164983 A164984

Keyword

nonn,tabl

Author

Mark Dols, Sep 03 2009

Extensions

Rows 10-11 from Michel Marcus, Feb 01 2023

Status

approved