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 A236076 A skewed version of triangular array A122075. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified July 19 17:03 EDT 2024. Contains 374410 sequences. (Running on oeis4.)