0,3

The k-th row contains 2k+1 numbers.

Columns in the right half consist of convolutions of the Lucas numbers with the natural numbers.

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.

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)}.

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.

....................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

........1,3,4,7,11,18,26,28,19,7,1

.....1,3,4,7,11,18,29,44,54,47,26,8,1

..1,3,4,7,11,18,29,47,73,98,101,73,34,9,1

(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)

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.

Adjacent sequences: A027957 A027958 A027959 this_sequence A027961 A027962 A027963

Sequence in context: A035690 A124794 A097560 this_sequence A131248 A116445 A110291

nonn,tabf

Clark Kimberling (ck6(AT)evansville.edu)

Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), May 04 2005