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a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.
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%I #7 Nov 23 2019 13:40:12

%S 1,1,1,5,11,7,43,85,19,341,683,455,2731,5461,3641,21845,43691,9709,

%T 174763,349525,233017,1398101,2796203,1864135,11184811,22369621,

%U 1657009,89478485,178956971,119304647,715827883,1431655765,954437177,5726623061,11453246123

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

%C a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)).

%e See Example in A327320.

%t 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 r = Sqrt[2]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];

%t Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]; (* A327320 *)

%t Table[f[x, n] /. x -> 0, {n, 1, 30}] (* A329005 *)

%t Table[f[x, n] /. x -> 1, {n, 1, 30}] (* A329006 *)

%t Table[f[x, n] /. x -> 2, {n, 1, 30}] (* A329007 *)

%t (* _Peter J. C. Moses_, Nov 01 2019 *)

%Y Cf. A327320, A329006, A329007.

%K nonn

%O 1,4

%A _Clark Kimberling_, Nov 08 2019