OFFSET
0,3
COMMENTS
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.
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..10000
FORMULA
T(n, k) = Lucas(k+1) for k <= n, otherwise the (2n-k)th coefficient of the power series for (1+2*x)/{(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 2 <= k <= 2*n-1.
EXAMPLE
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
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:
for n from 0 to 6 do for k from 0 to 2*n do print(T(n, k)); od:od: # Nathaniel Johnston, Apr 18 2011
MATHEMATICA
t[_, 0] = 1; t[_, 1] = 3; t[n_, k_] /; (k == 2*n) = 1; t[n_, k_] := t[n, k] = t[n-1, k-2] + t[n-1, k-1]; Table[t[n, k], {n, 0, 8}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Dec 27 2013 *)
PROG
(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)) /* Ralf Stephan, May 04 2005 */
(Sage)
def T(n, k):
if (k<0 or k>2*n): return 0
elif (k==0 or k==2*n): return 1
elif (k==1): return 3
else: return T(n-1, k-2) + T(n-1, k-1)
[[T(n, k) for k in (0..2*n)] for n in (0..12)] # G. C. Greubel, Jun 01 2019
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
EXTENSIONS
Edited by Ralf Stephan, May 04 2005
STATUS
approved