A027011 Triangular array T read by rows: T(n,k) = t(n, 2k+1) for 0 <= k <= floor((2n-1)/2), t given by A027960, n >= 0.

3, 3, 4, 3, 7, 5, 3, 7, 15, 6, 3, 7, 18, 28, 7, 3, 7, 18, 44, 47, 8, 3, 7, 18, 47, 98, 73, 9, 3, 7, 18, 47, 120, 199, 107, 10, 3, 7, 18, 47, 123, 291, 373, 150, 11, 3, 7, 18, 47, 123, 319, 661, 654, 203, 12, 3, 7, 18, 47, 123, 322, 806, 1404, 1085, 267, 13, 3, 7, 18, 47

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy
• tabl

Comments

Right-edge columns are polynomials approximating Lucas(2n).

Examples

			                                          3
                                     3,   4
                                3,   7,   5
                           3,   7,  15,   6
                      3,   7,  18,  28,   7
                 3,   7,  18,  44,  47,   8
            3,   7,  18,  47,  98,  73,   9
       3,   7,  18,  47, 120, 199, 107,  10
  3,   7,  18,  47, 123, 291, 373, 150,  11
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

This is a bisection of the "Lucas array " A027960, see A026998 for the other bisection.

Right-edge columns include A027965, A027967, A027969, A027971.

An earlier version of this entry had (unjustifiably) each row starting with 1.

Programs

Maple

t:=proc(n,k)option remember:if(k=0 or k=2*n)then return 1:elif(k=1)then return 3:else return t(n-1,k-2) + t(n-1,k-1):fi:end:
T:=proc(n,k)return t(n,2*k+1):end:
for n from 0 to 8 do for k from 0 to floor((2*n-1)/2) do print(T(n,k));od:od: # Nathaniel Johnston, Apr 18 2011

Mathematica

t[n_, k_] := t[n, k] = If[k == 0 || k == 2*n, 1, If[k == 1, 3, t[n-1, k-2] + t[n-1, k-1]]]; T[n_, k_] := t[n, 2*k+1]; Table[T[n, k], {n, 1, 12}, {k, 0, (2*n-1)/2}] // Flatten (* Jean-François Alcover, Nov 18 2013, after Nathaniel Johnston *)

Formula

T(n, k) = Lucas(2n) = A005248(n) for 2k+1 <= n, otherwise the (2n-2k+1)-th coefficient of the power series for (1+2x)/((1-x-x^2)(1-x)^(2k-n+1)).

Extensions

Edited by Ralf Stephan, May 05 2005