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A329005 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.

%I A329005 #7 Nov 23 2019 13:40:12
%S A329005 1,1,1,5,11,7,43,85,19,341,683,455,2731,5461,3641,21845,43691,9709,
%T A329005 174763,349525,233017,1398101,2796203,1864135,11184811,22369621,
%U A329005 1657009,89478485,178956971,119304647,715827883,1431655765,954437177,5726623061,11453246123
%N A329005 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.
%C A329005 a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)).
%e A329005 See Example in A327320.
%t A329005 c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@Variables /@ #1 &)[List @@ poly], 0], poly];
%t A329005 r = Sqrt[2]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
%t A329005 Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]];  (* A327320 *)
%t A329005 Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329005 *)
%t A329005 Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329006 *)
%t A329005 Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329007 *)
%t A329005 (* _Peter J. C. Moses_, Nov 01 2019 *)
%Y A329005 Cf. A327320, A329006, A329007.
%K A329005 nonn
%O A329005 1,4
%A A329005 _Clark Kimberling_, Nov 08 2019