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EXAMPLE
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a(0) = 6 because A083952(6) is an isolated 1,
a(1) = 0 because A083952(0) = A083952(2) = 1 but A083952(4) = 2,
a(2) = 198 because A083952(198) = A083952(200) = A083952(202) = 1, but A083952(196) = A083952(204) = 2, etc.
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MATHEMATICA
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a[n_] := a[n] = Block[{s = Sum[ a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; t = Union@ Table[ If[ a[n] == 1, n, 0], {n, 0, 5506}]; f[n_] := Block[{k = n, m = 2}, If[n == 1, 0, While[m < Length@t && t[[m - 1]] + 2 == t[[m]] || t[[m]] + 2k != t[[m + k]] || t[[m + k]] + 2 == t[[m + k + 1]], m++ ]; If[m < Length@t - k, t[[m]], 0]]]; Table[ f[n], {n, 0, 9}]
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