A236076 A skewed version of triangular array A122075.

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 1, 7, 8, 0, 0, 0, 4, 15, 13, 0, 0, 0, 1, 12, 30, 21, 0, 0, 0, 0, 5, 31, 58, 34, 0, 0, 0, 0, 1, 18, 73, 109, 55, 0, 0, 0, 0, 0, 6, 54, 162, 201, 89, 0, 0, 0, 0, 0, 1, 25, 145, 344, 365, 144, 0, 0, 0, 0, 0, 0, 7, 85, 361

Offset

0,3

Comments

Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Subtriangle of the triangle A122950.

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

H. Fuks and J.M.G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv:1306.1189 [nlin.CG], 2013.

Formula

G.f.: (1+x*y)/(1 - x*y - x^2*y - x^2*y^2).

T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k < 0 or if k > n.

Sum_{k=0..n} T(n,k) = 2^n = A000079(n).

Sum_{n>=k} T(n,k) = A078057(k) = A001333(k+1).

T(n,n) = Fibonacci(n+2) = A000045(n+2).

T(n+1,n) = A023610(n-1), n >= 1.

T(n+2,n) = A129707(n).

Example

Triangle begins:
  1;
  0,  2;
  0,  1,  3;
  0,  0,  3,  5;
  0,  0,  1,  7,  8;
  0,  0,  0,  4, 15, 13;
  0,  0,  0,  1, 12, 30, 21;
  0,  0,  0,  0,  5, 31, 58, 34;

Mathematica

T[n_, k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, 1, If[k==0, 0, If[n==1 && k==1, 2, T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]]]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 21 2019 *)

Prog

(Haskell)
a236076 n k = a236076_tabl !! n !! k
a236076_row n = a236076_tabl !! n
a236076_tabl = [1] : [0, 2] : f [1] [0, 2] where
   f us vs = ws : f vs ws where
     ws = [0] ++ zipWith (+) (zipWith (+) ([0] ++ us) (us ++ [0])) vs
-- Reinhard Zumkeller, Jan 19 2014
(PARI)
{T(n, k) = if(k<0 || k>n, 0, if(n==0 && k==0, 1, if(k==0, 0, if(n==1 && k==1, 2, T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2) ))))}; \\ G. C. Greubel, May 21 2019
(Sage)
def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    elif (k==0): return 0
    elif (n==1 and k==1): return 2
    else: return T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 21 2019

Crossrefs

Cf. variant: A055830, A122075, A122950, A208337.

Cf. A167704 (diagonal sums), A000079 (row sums).

Cf. A111006.

Sequence in context: A202502 A219839 A154312 * A364021 A363899 A119900

Adjacent sequences: A236073 A236074 A236075 * A236077 A236078 A236079

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Jan 19 2014

Status

approved