A027011 - OEIS

%I #22 Oct 07 2017 02:35:40

%S 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,

%T 7,18,47,120,199,107,10,3,7,18,47,123,291,373,150,11,3,7,18,47,123,

%U 319,661,654,203,12,3,7,18,47,123,322,806,1404,1085,267,13,3,7,18,47

%N 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.

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

%H Nathaniel Johnston, <a href="/A027011/b027011.txt">Table of n, a(n) for n = 1..10000</a>

%F 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)).

%e                                           3

%e                                      3,   4

%e                                 3,   7,   5

%e                            3,   7,  15,   6

%e                       3,   7,  18,  28,   7

%e                  3,   7,  18,  44,  47,   8

%e             3,   7,  18,  47,  98,  73,   9

%e        3,   7,  18,  47, 120, 199, 107,  10

%e   3,   7,  18,  47, 123, 291, 373, 150,  11

%p 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:

%p T:=proc(n,k)return t(n,2*k+1):end:

%p 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

%t 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_ *)

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

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

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

%K nonn,easy,tabl

%O 1,1

%A _Clark Kimberling_

%E Edited by _Ralf Stephan_, May 05 2005