A027960 'Lucas array': triangular array T read by rows.

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, 1, 3, 4, 7, 11, 18, 29, 47, 76, 120, 171, 199, 174, 107, 43, 10, 1

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

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.

Sequence in context: A206496 A097560 A218905 * A319182 A247282 A246685

Adjacent sequences: A027957 A027958 A027959 * A027961 A027962 A027963

Keyword

nonn,easy,tabf

Author

Clark Kimberling

Extensions

Edited by Ralf Stephan, May 04 2005

Status

approved