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Conjecture: a(n) = v_t_n for n > 0 with a(0) = 1 where we start with vector v of fixed length n m with elements v_i = 1 , then set t = v and for i=1..nm-1, for j=i+1..n m apply [v_i, v_j] := [v_i + 2*v_j, 2*v_i + v_j] (here square brackets mean that instead of sequentially assigning v_i and then v_j, we reserve their values (for example, as A = v_i, B = v_j) and then assign them in any order) and t_{i+1} := v_{i+1} (after ending each cycle for j). Changing It also looks like that if we change 2*v_i + v_j to xz*v_i + v_j leads it gives us to a(n+1) = Sum_{k=0..n} A090981(n, k)*2^(n-k) for n >= 0. - Mikhail Kurkov, Aug 14 2024

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Conjecture: a(n) = v_n for n > 0 with a(0) = 1 where we start with vector v of length n with elements v_i = 1 and for i=1..n-1, for j=i+1..n apply [v_i, v_j] = [v_i + 2*v_j, 2*v_i + v_j] (here square brackets mean that instead of sequentially assigning v_i and then v_j, we reserve their values (for example, as A = v_i, B = v_j) and then assign them in any order). It also looks like Changing 2*v_i + v_j to x*v_i + v_j leads us to a(n+1) = 1 + 2*Sum_{ik=10..n} v_i A090981(n, k)*2^(n-k) for n > = 0. - Mikhail Kurkov, Aug 14 2024

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Conjecture: a(n) = v_n for n > 0 with a(0) = 1 where we start with vector v of length n with elements v_i = 1 and for i=1..n-1, for j=i+1..n apply [v_i, v_j] = [v_i + 2*v_j, 2*v_i + v_j] (here square brackets mean that instead of sequentially assigning v_i and then v_j, we reserve their values (for example, as A = v_i, B = v_j) and then assign them in any order). It also looks like a(n+1) = 1 + 2*Sum_{i=1..n} v_i for n > 0. - Mikhail Kurkov, Aug 14 2024

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(MAGMAMagma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (5-Sqrt((1-9*x)/(1-x)))/4 )); // G. C. Greubel, Feb 07 2019

(MAGMAMagma) a:=[1, 3]; for n in [3..21] do Append(~a, (2*(-7+5*n)*a[n-1] + 9*(2-n)*a[n-2]) div n); end for ; [1] cat a; // Marius A. Burtea, Jan 20 2020

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reviewed