%I A027960
%S A027960 1,1,3,1,1,3,4,4,1,1,3,4,7,8,5,1,1,3,4,7,11,15,13,6,1,1,3,4,7,11,18,26,
%T A027960 28,19,7,1,1,3,4,7,11,18,29,44,54,47,26,8,1,1,3,4,7,11,18,29,47,73,98,
%U A027960 101,73,34,9,1,1,3,4,7,11,18,29,47,76,120,171
%N A027960 'Lucas array': triangular array T read by rows.
%C A027960 The k-th row contains 2k+1 numbers.
%C A027960 Columns in the right half consist of convolutions of the Lucas numbers 
               with the natural numbers.
%C A027960 T(n,k) = number of strings s(0),...,s(n) such that s(n)=n-k. s(0) in 
               {0,1,2}, s(1)=1 if s(0) in {1,2}, s(1) in {0,1,2} if s(0)=0 and for 
               1<=i<=n, s(i)=s(i-1)+d, with d in {0,2} if s(i)=2i, in {0,1,2} if 
               s(i)=2i-1, in {0,1} if 0<=s(i)<=2i-2.
%F A027960 T(n, k) = Lucas(k+1) for k<=n, otherwise the (2n-k)th coefficient of 
               the power series for (1+2x)/{(1-x-x^2)(1-x)^(k-n)}.
%F A027960 Recurrence: T(n, 0)=T(n, 2n)=1 for n >= 0; T(n, 1)=3 for n >= 1; and 
               for n >= 2, T(n, k)=T(n-1, k-2)+T(n-1, k-1) for k=2, 3, ..., 2n-1.
%e A027960 ....................1
%e A027960 ..................1,3,1
%e A027960 ................1,3,4,4,1
%e A027960 ..............1,3,4,7,8,5,1
%e A027960 ...........1,3,4,7,11,15,13,6,1
%e A027960 ........1,3,4,7,11,18,26,28,19,7,1
%e A027960 .....1,3,4,7,11,18,29,44,54,47,26,8,1
%e A027960 ..1,3,4,7,11,18,29,47,73,98,101,73,34,9,1
%o A027960 (PARI) T(r,n)=if(r<0||n>2*r,return(0));if(n==0||n==2*r,return(1));if(n==1,
               3,T(r-1,n-1)+T(r-1,n-2)) (from R. Stephan)
%Y A027960 Central column is the Lucas numbers without initial 2, cf. A000204. Row 
               sums are A036563. Columns in the right half include A027961, A027962, 
               A027963, A027964, A053298. Bisection triangles are in A026998 and 
               A027011.
%Y A027960 Sequence in context: A035690 A124794 A097560 this_sequence A131248 A116445 
               A110291
%Y A027960 Adjacent sequences: A027957 A027958 A027959 this_sequence A027961 A027962 
               A027963
%K A027960 nonn,tabf
%O A027960 0,3
%A A027960 Clark Kimberling (ck6(AT)evansville.edu)
%E A027960 Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), May 04 2005