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%I #5 Feb 16 2022 04:15:42
%S 1,1,1,1,1,1,1,3,9,27,81,243,729,2189,6601,20141,63009,205989,718905,
%T 2720543,11183601,49321367,228895201,1097860903,5371546897,
%U 26598018425,132755261681,667027581401,3376011676481,17249045903945,89270689572497,470069622480667
%N G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - 2*x)) / (1 - 2*x).
%C Shifts 6 places left under 2nd-order binomial transform.
%F a(0) = ... = a(5) = 1; a(n) = Sum_{k=0..n-6} binomial(n-6,k) * 2^k * a(n-k-6).
%t nmax = 31; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[n_] := a[n] = If[n < 6, 1, Sum[Binomial[n - 6, k] 2^k a[n - k - 6], {k, 0, n - 6}]]; Table[a[n], {n, 0, 31}]
%Y Cf. A004211, A007472, A010750, A210543, A275936, A351342, A351343, A351344
%K nonn
%O 0,8
%A _Ilya Gutkovskiy_, Feb 08 2022
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